DT–OPERATORS AND DECOMPOSABILITY OF VOICULESCU’S CIRCULAR OPERATOR

KEN DYKEMA AND UFFE HAAGERUP

Abstract. The DT–operators are introduced, one for every pair (μ, c) consisting of a compactly supported Borel probability measure μ on the complex plane and a constant c > 0. These are operators on Hilbert space that are defined as limits in ∗– moments of certain upper triangular random matrices. The DT–operators include Voiculescu’s circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT–operator is strongly decomposable. We also show that a DT–operator generates a II_{1}–factor, whose isomorphism class depends only on the number and sizes of atoms of μ. Those DT–operators that are also R–diagonal are identified. For a quasi–nilpotent DT–operator T, we find the distribution of T^{∗}T and a recursion formula for general ∗–moments of T.

Contents

1. Introduction 2 1.1. Local spectral theory and decomposability 2 1.2. DT–operators 3 2. The DT ∗–moment distributions 5 3. Examples of DT–elements 16 4. DT–operators in finite von Neumann algebras 21 5. Decomposability of DT–operators 30 6. Von Neumann algebras generated by DT–operators 34 7. DT–operators that are also R–diagonal 36 8. The distribution of T^{∗}T 39 9. ∗–moments of the operator T 50 References 57

Date: 22 April, 2002. K.D. supported in part by NSF grant DMS–0070558. U.H. affiliated with MaPhySto, Centre

for Mathematical Physics and Stochastics, which is funded by a grant from The Danish National Research Foundation. Both authors thank also the Mathematical Sciences Research Institute in Berkeley, where part of this work was carried out in the spring of 2001. U.H. was employed by the Clay Mathematical Institute as a Clay Research Scholar while he was at MSRI. Research at MSRI is supported in part by NSF grant DMS–9701755.

1. Introduction

1.1. Local spectral theory and decomposability. Let us begin by describing some ideas and results in local spectral theory related to decomposability of operators. Our exposition is drawn from the book [16] by Laursen and Neumann, and depends on work of Foiaş [8], [9], Apostol [2], [3], Albrecht [1], Jafarian and Vasilescu [14] and others (see [16] for some history, more detailed citations and proofs). Theorem 1.1 below and the considerations surrounding it apply to bounded operators on Banach spaces, but in keeping with the subject of this paper we will restrict to discussion of a bounded operator T on a Hilbert space ℋ.

Such an operator T is said to be decomposable if, for every cover C = U ∪ V of the complex plane by two open subsets U and V, there are T –invariant closed subspaces ℋ′ and ℋ′′ of ℋ such that the spectra of the restrictions of T satisfy σ(T ↾_{ℋ′} ) ⊆ U and σ(T ↾_{ℋ′′} ) ⊆ V, and such that ℋ = ℋ′ + ℋ′′.

Given a bounded operator T on ℋ, a spectral capacity for T is a mapping E from the set of all closed subsets of C into the set of all closed T –invariant subspaces of ℋ, such that

(i) E(∅) = {0} and E(ℋ) = C,

(ii) ℋ = E(\overline{U_{1}}) + ⋯ + E(\overline{U_{n}}) = ℋ for every open cover \{U_{1}, …, U_{n}\} of C,

(iii) E(⋂_{n=1}^{∞} F_{n}) = ⋂_{n=1}^{∞} E(F_{n}) for any closed subsets F_{1}, F_{2}, … of C,

(iv) σ(T ↾_{E(F)}) ⊆ F for every closed subset F of C, (with the convention that the operator on the Hilbert space {0} has empty spectrum).

If ξ ∈ ℋ, the local resolvent set, ρ_{T} (ξ), of T at ξ is the union of all open subsets U of C for which there exist holomorphic vector–valued functions f_{U} : U → ℋ such that (T −λ)f_{U}(λ) = ξ for all λ ∈ U. The local spectrum of T at ξ is then defined to be σ_{T} (ξ) = C\ρ_{T} (ξ). For any subset A of C, the corresponding local spectral subspace of T is

ℋ_{T} (A) = {ξ ∈ ℋ | σ_{T} (ξ) ⊆ A}.

It is clear that ℋ_{T} (A) is T –invariant, and even T –hyperinvariant, namely invariant under every operator that commutes with T. Hence letting p_{T} (A) deonte the projection of ℋ onto the closure of ℋ_{T} (A), it follows that p_{T} (A) lies in the von Neumann algebra vN(T) that is generated by T. In fact, by using standard techniques like those in the proofs of Lemmas 2.3 and 2.4 of [6], one can show that the projection p_{T} (A), as an element of the abstract W^{∗}–algebra M that is isomorphic to vN(T), is independent of the particular representation of M on a Hilbert space. (Our convention is that a von Neumann algebra is the image of a normal representation of an abstract W^{∗}–algebra on a Hilbert space.)

Theorem 1.1. (cf. [16, §1.2]). Let T be a bounded operator on a Hilbert space ℋ. Then the following are equivalent:

(i) T is decomposable,

(ii) T has a spectral capacity, (iii) for every closed subset F of C, ℋ_{T} (F) is closed and

σ((1 − p_{T} (F))T) ⊆ \overline{σ(T)\F} ,

where the spectrum is of (1 − p_{T} (F))T considered as an operator on ℋ_{T} (F)^{⊥}. Moreover, if T is decomposable then the map F ↦ ℋ_{T} (F) is the unique spectral capacity for T.

An operator T is said to be strongly decomposable if it is decomposable and, morover, if the restriction T ↾_{ℋ_{T} (F)} is decomposible for every closed subset F of C. In light of the above comments about p_{T} (A) in abstract W^{∗}–algebras, it is clear that decomposability and strong decomposability of operators can be thought of as algebraic properties of elements of W^{∗}–algebras.

1.2. DT–operators. In this paper, we introduce the DT–operators. These are operators on Hilbert space that are defined as limits of certain large random matrices and are intimately related to free probability theory, (see the book [27]). We will introduce them first in the context of ∗–noncommutative probability spaces. A ∗– noncommutative probability space is a pair (𝔄, τ), where 𝔄 is a unital algebra over C with an involution usually denoted a ↦ a^{∗}, and where τ : A → C is a linear functional that is unital (τ(1) = 1) and positive (τ(a^{∗}a) ≥ 0). A ∗–noncommutative probability space is said to be tracial if τ satisfies the trace property. An important example is the tracial ∗–noncommutative probability space of n×n random matrices, (ℳ_{n}, τ_{n}), where ℳ_{n} is an algebra of n×n matrices whose entries are random variables over a classical probability space having moments of all orders and where, viewing an element a ∈ ℳ_{n} as an M_{n}(C)–valued random variable, τ_{n}(a) is the expectation of tr_{n}(a), where tr_{n} is the normalized trace on M_{n}(C). Let T_{n} ∈ ℳ_{n} be a strictly upper triangular random matrix, the real and imaginary parts of whose entries above the diagonal form a family of n(n−1) i.i.d. gaussian random variables, each having mean 0 and variance 1/2n. Fix a Borel measure μ on C having compact support, and let D_{n} ∈ ℳ_{n} be a diagonal random matrix having mutually independent μ–distributed diagonal entries and such that, as matrix–valued random variables, D_{n} and T_{n} are independent. In §2, we show that the pair D_{n}, T_{n} converges in joint ∗–moments as n → ∞. This means that τ_{n} of any fixed word in D_{n}, D_{n}^{∗}, T_{n} and T_{n}^{∗} converges as n → ∞; in fact, we have a combinatorial formula involving non–crossing pairings giving the limiting value. In particular, taking Z_{n} = D_{n} + cT_{n} for any fixed c > 0, it follows that Z_{n} converges in ∗–moments as n → ∞. This means that for every k ∈ N and all ϵ(1), …, ϵ(k) ∈ {∗, 1}, the limit

lim_{n→∞} τ_{n}(Z_{n}^{ϵ(1)}Z_{n}^{ϵ(2)} ⋯ Z_{n}^{ϵ(k)})  (1)

exists. We define a DT(μ, c)–element to be an element Z ∈ 𝔄 of a ∗–noncommutative probability space (𝔄, τ) whose ∗–moments τ(Z^{ϵ(1)} ⋯ Z^{ϵ(k)}) are given by the limits (1). An element is a DT–element if it is a DT(μ, c)–element for some μ and c.

Drawing on results from [6], we have (Theorem 2.13) that the diagonal part D_{n} of Z_{n} = D_{n} + cT_{n} can be modified quite extensively from the situation described above without changing that Z_{n} converges in ∗–moments to the DT(μ, c)–element Z.

This flexibility in the random matrix model for Z is important in our investigation of DT–elements.

In §3, we will show that the class of DT–elements includes some elements arising naturally in free probability theory, namely Voiculsecu’s circular element, the circular free Poisson elements, and the so called elliptic elements, which are deformations of the circular element.

In §4, we construct DT–elements as operators on Hilbert space; more specifically, for every μ and c, we give a canonical construction of a DT(μ, c)–element in the ∗–noncommutative probability space (L(F_{2}), τ), where L(F_{2}) ⊆ B(ℓ^{2}(F_{2})) is the von Neumann algebra generated by the left regular representation of the nonabelian free group on two generators and τ is the canonical tracial state. A W^{∗}–noncommutative probability space is a ∗–noncommutative probability space (M, τ), where M is a von Neumann algebra and τ is a normal state. We say that a DT(μ, c)–operator is a DT(μ, c)–element in W^{∗}–noncommutative probability space (M, τ), where the state τ is assumed to be faithful. Of course, an element of a von Neumann algebra with a specified faithful normal state is called a DT–operator if it is a DT(μ, c)–operator for some μ and c.

Using the flexibility in the random matrix model for a DT–element and our construction of DT–operators in L(F_{2}), we prove (Theorem 4.12) that if

A = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1N} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{N−1,N} \\ 0 & ⋯ & 0 & a_{N} \end{matrix} )  (2)

is an upper triangular N × N matrix of operators, whose entries are mutually ∗–free, where each b_{ij} is a circular element and where a_{j} is a DT(μ_{j}, \frac{c}{\sqrt{N}})–operator, then A is itself a DT(μ, c)–operator, where μ is the average of μ_{1}, …, μ_{N}. This result appears as a basic tool in our proofs of a number of interesting facts about DT–operators.

In §5, we use the upper triangular picture (2) to investigate the local spectral subspaces of DT–operators. We show that if Z is a DT(μ, c)–operator in a W^{∗}– noncommutative probability space (M, τ), then the spectrum of Z is equal to the support of μ. Moreover, if B is any Borel set in C, then the projection p_{Z}(B) onto the closure of the spectral subspace of B has trace τ(p_{Z}(B)) = μ(B). Also, if 0 < μ(B) < 1, then Zp_{Z}(B) and (1 − p_{Z}(B))Z are DT–operators corresponding to the restrictions of μ to B and to its complement B^{c}, respectively. From these facts, we show that the Brown measure of Z is μ and Z is strongly decomposable.

Since the projections p_{Z}(B) belong to the von Neumann algebra generated by Z, we have proved, so long as the support of μ has more than one point, that Z has nontrivial invariant subspaces affiliated to the von Neumann algebra it generates. On the other hand, let T be a DT(δ_{0}, 1)–operator. Then T is a natural candidate for an operator without any invariant subspaces relative to the von Neumann algebra it generates. This issue is related to questions about the von Neumann algebra generated by the canonical copy of T that we construct in L(F_{2}). In §6, we investigate this von Neumann algebra, and prove it is an irreducible subfactor of L(F_{2}).

The circular free Poisson operators were introduced in [6]. In that paper, we showed that circular free Poisson operators have nontrivial invariant subspaces relative to the von Neumann algebras they generate. As remarked above, every circular free Poisson operator is a DT–operator, and it is clear that the invariant subspaces constructed in [6] are the local spectral subspaces for annular subsets of the spectrum. The circular free Poisson operators are also examples of R–diagonal operators, in the sense of Nica and Speicher [19]. Śniady and Speicher [21] have extended the results of [6] in another direction, showing that every R–diagonal operator has an increasing, one–parameter family of nontrivial invariant subspaces relative to its von Neumann algebra. In §7, we show that the only DT–operators that are also R-diagonal operators are the circular free Poisson operators. Hence, the overlap between the results of [21] and of this paper lies in their common antecedent [6].

In §8, we find the moments of T^{∗}T:

τ((T^{∗}T)^{n}) = \frac{n^{n}}{(n + 1)!}.

We show that these are the moments of a probability distribution supported on [0, e] which is Lebesgue absolutely continuous, and we explicitly describe its density function. As corollaries, we have ∥T ∥ = \sqrt{e} and ker T = \{0\}.

In §9, we prove a recursion formula for general ∗–moments

τ((T^{∗})^{k_{1}}T^{ℓ_{1}} … (T^{∗})^{k_{n}}T^{ℓ_{n}})

of T. We conclude by conjecturing a pretty formula for the special ∗–moments

τ(((T^{∗})^{k}T^{k})^{n}),

which has been proved in some cases and checked by computation in others.

2. The DT ∗–moment distributions

In this section, we show convergence in ∗–moments of certain sequences of upper triangular random matrices and we use them to define DT–elements. We also prove some preliminary results about DT–elements. The random matrix results seem to be related to those in [20], but go in a rather different direction.

As in the introduction, we will let (ℳ_{n}, τ_{n}) be the ∗–noncommutative probability space of n×n random matrices, over a (large enough) classical probability space. We will use the notations GRM, UTGRM, HURM and SGRM for special sorts of random matrices in ℳ_{n}, as described in Notations 3.1 and 4.1 of [6]. In particular, we say that a random variable a ∈ ℒ is complex N(0, σ^{2}) if Re a and Im a are independent real gaussian random variables each having first moment 0 and second moment σ^{2}/2, and for T ∈ ℳ_{n}, we have T ∈ UTGRM(n, σ^{2}) (the acronym is for upper triangular gaussian random matrix) if the entries t_{ij} of T (1 ≤ i, j ≤ n) satisfy that t_{ij} = 0 whenever i ≥ j and that (t_{ij})_{1≤i<j≤n} is an independent family of random variables, each of which is complex N(0, σ^{2}).

Theorem 2.1. Let μ be a compactly supported Borel probability measure on C. For n ∈ N, let D_{n} be a diagonal random matrix whose diagonal entries are i.i.d. μ– distributed random variables and let T_{n} ∈ UTGRM(n, \frac{1}{n}) be such that D_{n} and T_{n} are independent (as matrix–valued random variables). Then the pair D_{n}, T_{n} converges jointly in ∗–moments as n → ∞.

The proof will come later in this section, but we now use the result to define DT–elements.

Definition 2.2. Let μ be a compactly supported Borel probability measure on C and let c > 0. Let D_{n} and T_{n} be as in Theorem 2.1 above, for this choice of μ. An element z of a ∗–noncommutative probability space (A, ϕ) is called a DT(μ, c)–element if its ∗–moment distribution is the limit ∗–moment distribution of D_{n} + cT_{n} as n → ∞, i.e. if

ϕ(z^{ϵ(1)}z^{ϵ(2)} ⋯ z^{ϵ(k)}) = lim_{n→∞} τ_{n}(Z_{n}^{ϵ(1)}Z_{n}^{ϵ(2)} ⋯ Z_{n}^{ϵ(k)})

for every k ∈ N and ϵ(1), …, ϵ(k) ∈ {1, ∗}, where Z_{n} = D_{n} + cT_{n}.

An element of a ∗–noncommutative probability space is called a DT–element if it is a DT(μ, c)–element for some μ and c > 0.

We note that the letters “DT” signify diagonal + (gaussian upper) triangular.

Remark 2.3. We could use the Gelfand–Naimark–Segal representation to show, at the same time we prove Theorem 2.1, that every DT–element can be realized as a bounded operator; more precisely, for every μ and c there is a von Neumann algebra M equipped with a normal tracial state τ such that there is a DT(μ, c)–element in the ∗–noncommutative probability space (M, τ). However, we will postpone a proof of this to §4, where we will give a construction of an arbitrary DT–element in the free group factor (L(F_{2}), τ); see Theorem 4.4 and Remark 4.8.

Our proof of Theorem 2.1 begins with a combinatorial analysis of the limiting ∗–moments of T_{n}.

Recall that a pairing of {1, 2, …, k}, (for k even) is

σ = {{i_{1}, j_{i}}, {i_{2}, j_{2}}, …, {i_{k/2}, j_{k/2}}}

where {i_{1}, …, i_{k/2}, j_{1}, …, j_{k/2}} = {1, …, k}; a pairing σ is said to be crossing if i_{1} < i_{2} < j_{1} < j_{2} for some {i_{1}, j_{1}} ∈ σ and {i_{2}, j_{2}} ∈ σ and is said to be non–crossing otherwise.

Lemma 2.4. For every n ∈ N let T_{n} ∈ UTGRM(n, 1/n), Let k ∈ N and ϵ(1), …, ϵ(k) ∈ {∗, 1}. Then the limit

lim_{n→∞} τ_{n}(T_{n}^{ϵ(1)}T_{n}^{ϵ(2)} ⋯ T_{n}^{ϵ(k)})  (3)

exists. The value of the limit (3) is zero unless there is a non–crossing pairing σ of {1, 2, …, k} such that whenever {i, j} ∈ σ we have ϵ(i) ≠ ϵ(j); we will call such a non–crossing pairing σ compatible with ϵ(1), …, ϵ(k). The value of the limit (3) is

\frac{1}{(\frac{k}{2} + 1)!} ∑_{σ∈NCP(k) ; compatible} NTO(σ; ϵ(1), …, ϵ(k)),  (4)

where the sum is over all non–crossing pairings σ of {1, 2, …, k} that are compatible with ϵ(1), …, ϵ(k), and where NTO(σ; ϵ(1), …, ϵ(k)) is the positive integer obtained via the algorithm below.

Algorithm 2.5. Suppose k ∈ N and σ = {{i_{1}, j_{i}}, {i_{2}, j_{2}}, …, {i_{k/2}, j_{k/2}}} is a pairing of {1, 2, …, k} and let ϵ(1), …, ϵ(k) ∈ {∗, 1} be such that ϵ(i) ≠ ϵ(j) for every {i, j} ∈ σ. Then the integer NTO(σ; ϵ(1), …, ϵ(k)) is determined by the following algorithm.

(A). Let G be the k–gon graph with consecutive vertices v_{1}, …, v_{k} and consecutive edges e_{1}, …, e_{k}, with edge e_{j} having vertices v_{j} and v_{j+1}; (here and throughout the algorithm we take indices of vertices modulo k). Orient each edge e_{j} negatively (i.e. with arrow pointing from v_{j+1} to v_{j}) if ϵ(j) = 1 and positively (i.e. with arrow pointing from v_{j} to v_{j+1}) if ϵ(j) = ∗.

(B). Let Q = Q(σ; ϵ(1), …, ϵ(k)) be the quotient graph of G obtained by identifying edges e_{i} and e_{j} whenever {i, j} ∈ σ, in such a way that the orientations given the edges in part (A) are matched. Thus vertex v_{i} is identified with v_{j+1} and vertex v_{i+1} is identified with v_{j}. The edges of Q are given the orientations (i.e. arrows) inherited from the edges of G. Note that taken without orientation of its edges, Q is the quotient graph considered by Voiculescu in [25].

(C). Set NTO(σ; ϵ(1), …, ϵ(k)) = 0 if σ is a crossing pairing. If σ is a non–crossing pairing then by Lemma 2.9, Q is a tree and has \frac{k}{2} +1 vertices; let w_{1}, w_{2}, …, w_{\frac{k}{2} +1} be the vertices of Q. Consider the relation ← on \{w_{1}, w_{2}, …, w_{\frac{k}{2} +1}\} defined by w_{i} ← w_{j} if there is an edge in Q with vertices w_{i} and w_{j} and whose orientation is an arrow pointing to w_{i} from w_{j}. Let NTO(σ; ϵ(1), …, ϵ(k)) be the number of different total orderings < of \{w_{1}, w_{2}, …, w_{\frac{k}{2} +1}\} that extend the relation ←. Because Q is a tree it is clear that the transitive relation generated by ← is a partial ordering and hence that NTO(σ; ϵ(1), …, ϵ(k)) ≥ 1.

Example 2.6. We will use Algorithm 2.5 to find NTO(σ; ∗, 1, ∗, 1, ∗, 1, 1, ∗, 1, ∗) when σ = {{1, 6}, {2, 3}, {4, 5}, {7, 10}, {8, 9}}. Performing parts (A) and (B) yields

[figure: e_{1}; e_{10}; and; w_{1}; w_{2}; w_{3}; w_{4}; w_{5}; w_{6}]

for G and Q, respectively. The relation ← can be drawn schematically as

[figure: w_{1}; w_{2}; w_{3}; w_{4}; w_{5}; w_{6}]

and we count the number of total orderings extending ← to be

NTO(σ; ∗, 1, ∗, 1, ∗, 1, 1, ∗, 1, ∗) = 2 · 4 · 5 + 3 · 4 = 52.

The term 2 · 4 · 5 is for when w_{1} is placed above w_{4} and 3 · 4 is for when w_{1} is placed below w_{4}.

Given a pairing σ of {1, …, k}, we say that σ has paired neighbors if {i, i + 1} ∈ σ for some i ∈ {1, 2, …, k − 1}. In that case, σ − {i, i + 1} will denote the pairing of {1, …, k − 2} obtained from σ\{{i, i + 1}} by applying the map

j ↦ \begin{cases}j & if j < i \\ j − 2 & if j > i + 1. \end{cases}

We say that σ−{i, i+1} is the pairing obtained by removing paired neighbors {i, i+1} from σ. For example, if σ is as in Example 2.6 then

σ − {4, 5} = {{1, 4}, {2, 3}, {5, 8}, {6, 7}}.

The following lemma is well known; however we include a short proof for convenience.

Lemma 2.7. A pairing σ of {1, …, k} is non–crossing if and only if it can be reduced to the empty set by successively removing paired neighbors.

Proof. It is easy to see by induction that every non–crossing pairing has paired neighbors. Indeed, if σ is non–crossing and if {i, j} ∈ σ with i < j then the restriction of σ to {i + 1, i + 2, …, j − 1} is non-crossing. Moreover, if a pairing σ has paired neighbors {i, i + 1} then σ − {i, i + 1} is crossing if and only if σ is crossing.

Now the statement of the lemma is easily proved by induction on k/2. The case k = 2 is trivial. Let k ≥ 4. If σ has no paired neighbors then it cannot be reduced at all by removing paired neighbors, and σ is also crossing. If σ has paired neighbors {i, i + 1} then σ is non–crossing if and only if σ − {i, i + 1} is non–crossing, and by the induction hypothesis σ − {i, i + 1} can be reduced to the empty set by removing paired neighbors if and only if it is non–crossing. □

Remark 2.8. The above proof actually shows that every non–crossing pairing can be reduced to the empty set by removing paired neighbors and at every stage choosing arbitrary paired neighbors to be removed.

The following lemma is also well known. Again, for convenience we include a proof.

Lemma 2.9. Let σ be a pairing of {1, …, k} and let ϵ(1), …, ϵ(k) ∈ {∗, 1} be such that ϵ(i) ≠ ϵ(j) whenver {i, j} ∈ σ, and let Q = Q(σ; ϵ(1), …, ϵ(k)) be the quotient graph obtained in step (B) of Algorithm 2.5. If σ is non–crossing then Q is a tree having \frac{k}{2} + 1 vertices, while if σ is crossing then Q is not a tree and has ≤ \frac{k}{2} vertices.

Proof. If σ has no paired neighbors then every edge of the k–gon graph G is identified with another edge not its neighbor. Hence every vertex of G gets identified with at least one other vertex and thus Q has \frac{k}{2} edges and ≤ \frac{k}{2} vertices, and is therefore not a tree. If σ has paired neighbors {i, i + 1} then identifying edges e_{i} and e_{i+1} of G we get a (k − 2)–gon graph with a tail consisting of one edge:

[figure]

(where we have omitted to indicate the edges’ orientations). Then Q is obtained from the above graph by identifying edges of the inner (k − 2)–gon graph according to the pairing σ − {i, i + 1}. Continuing in this way to identify neighboring edges as many times as possible, after p steps, 1 ≤ p ≤ \frac{k}{2} − 2, we will have an inner (k − 2p)–gon with adjoined tails, each consisting of one or more trees, the tails together having p additional vertices. If σ is crossing then by Lemma 2.7 for some p the inner (k − 2p)– gon will have no neighboring edges identified, and thus each of its k − 2p vertices will be paired with at least one other vertex. The graph Q will then have at most \frac{k}{2} vertices, and Q will not be a tree. If σ is non–crossing then continuing until p = \frac{k}{2} −2 the graph will be

[figure: T_{4}; T_{3}; T_{2}; T_{1}]

where each of T_{1}, T_{2}, T_{3} and T_{4} represents one or more trees adjoined for a total of \frac{k}{2} − 2 additional vertices. The pairing of the remaining four edges of the inner 4–gon will be non–crossing, i.e.

[figure: or]

giving rise to the trees

[figure: T_{4}; T_{3}; T_{2}; T_{1}; or; T_{4}; T_{3}; T_{2}; T_{1}]

respectively, each of which has \frac{k}{2} + 1 vertices. □

Proof of Lemma 2.4. Let t(i, j, ϵ; n) denote the ijth entry of T_{n}^{ϵ}. Thus

(t(i, j, 1; n))_{1≤i<j≤n}  (5)

is a family of i.i.d. complex N(0, \frac{1}{n})–random variables, while t(i, j, 1; n) = 0 if i ≥ j and

t(i, j, ∗; n) = \overline{t(j, i, 1; n)}.

We have

τ_{n}(T_{n}^{ϵ(1)}T_{n}^{ϵ(2)} ⋯ T_{n}^{ϵ(k)}) =

= n^{−1} ∑_{i_{1},...,i_{k}∈{1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n)t(i_{2}, i_{3}, ϵ(2); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)),

where i_{k+1} means i_{1}. For a given choice of i_{1}, …, i_{k} ∈ {1, …, n} the quantity

E(t(i_{1}, i_{2}, ϵ(1); n)t(i_{2}, i_{3}, ϵ(2); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n))

is zero unless the k random variables t(i_{j}, i_{j+1}, ϵ(j); n), 1 ≤ j ≤ k, can be paired off so that each pair consists of mutually conjugate random variables. This happens if and only if there is a pairing σ of {1, …, k} that is compatible with ϵ(1), …, ϵ(k) and such that

i_{p} = i_{q+1} and i_{p+1} = i_{q} whenever {p, q} ∈ σ.  (6)

In this case, let Q = Q(σ; ϵ(1), …, ϵ(k)) be the quotient graph of the k–gon graph G as described in Algorithm 2.5. Label the jth vertex v_{j} of G with the value of i_{j}. Then condition (6) is equivalent to the condition that vertices of G that are sent by the quotient mapping to the same vertex in Q are labeled with the same value. Thus choosing i_{1}, …, i_{k} ∈ {1, …, n} so that (6) holds is equivalent to labeling the vertices of Q with values from {1, …, n}.

For a given choice of values of i_{1}, …, i_{k} there may be more than one pairing σ of {1, …, k} that is compatible with ϵ(1), …, ϵ(k) and so that (6) holds. However, if for some non–crossing pairing σ, the values of i_{1}, …, i_{k} are such that the corresponding labeling of the vertices of Q is with distinct values from {1, …, n}, then σ is the unique pairing of {1, …, k} such that (6) holds. Indeed, since Q, being a tree, has at most one edge connecting any two vertices, the different edges of Q are distinguished by the labels of their vertices; since Q is formed by pairing the edges of G, in the vertex labeling i_{1}, …, i_{k} of G, each edge of G must have exactly one other edge with the same set of vertex labels. Thus any choice of i_{1}, …, i_{n} yielding a distinct vertex labeling of the tree Q determines a unique pairing σ of the edges of G.

An upper bound for

τ_{n}(T_{n}^{ϵ(1)}T_{n}^{ϵ(2)} ⋯ T_{n}^{ϵ(k)})  (7)

is

n^{−1} ∑_{σ∈P(k) ; compatible} ∑_{vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)),  (8)

where the first sum is over all pairings σ of {1, …, k} that are compatible with ϵ(1), …, ϵ(k) and the second sum is over all vertex labelings of the quotient graph Q = Q(σ; ϵ(1), …, ϵ(k)) with elements of {1, …, n}; then i_{1}, …, i_{k} appearing in (8) are assigned values based on the vertex labeling of Q by giving i_{j} the value of the label of the vertex of Q to which the vertex v_{j} in G is mapped under the quotient mapping. A lower bound for (7) is

n^{−1} ∑_{σ∈NCP(k) ; compatible} ∑_{distinct vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)),  (9)

where the first sum is over all non–crossing pairings σ of {1, …, k} that are compatible with ϵ(1), …, ϵ(k) and the second sum is over all labelings of the \frac{k}{2} + 1 vertices of Q(σ; ϵ(1), …, ϵ(k)) with distinct elements from {1, …, n}, the corresponding values of i_{1}, …, i_{k} in (9) being determined as described above. The difference between the upper bound (8) and the lower bound (9) is

n^{−1} ∑_{σ∈NCP(k) ; compatible} ∑_{non–distinct vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n))

+n^{−1} ∑_{σ∈P(k)\ NCP(k) compatible} ∑_{vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)),

(10)

where σ ∈ P(k)\ NCP(k) means that σ is a crossing pairing of {1, …, k}. There is a constant C_{k} depending only on k such that for all n ∈ N and all i_{1}, …, i_{k} ∈ {1, …, n},

0 ≤ E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)) ≤ C_{k}n^{−k/2}.

For a given σ ∈ NCP(k), the number of different non–distinct vertex labelings of Q(σ; ϵ(1), …, ϵ(k) is bounded above by \binom{\frac{k}{2} +1}{2}n^{k/2}. Hence

0 ≤ ∑_{non–distinct vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from \{1,...,n\}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)) ≤ C_{k}\binom{\frac{k}{2} + 1}{2}.

For a given σ ∈ P(k)\ NCP(k), since by Lemma 2.9 Q = Q(σ; ϵ(1), …, ϵ(k)) has ≤ \frac{k}{2} vertices, the number of different vertex labelings of Q is bounded above by n^{k/2} and therefore

0 ≤ ∑_{vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)) ≤ C_{k} .

Thus we see that the difference (10) tends to zero as n → ∞, and the limit as n → ∞ of (7) will equal the limit as n → ∞ of the lower bound (9), if the latter exists.

We will now show that the lower bound (9) converges as n → ∞ to the desired value (4). Fix σ ∈ NCP(k) compatible with ϵ(1), …, ϵ(k), fix a distinct vertex labeling of Q = Q(σ; ϵ(1), …, ϵ(k)) from {1, …, n} and let i_{1}, …, i_{k} ∈ {1, …, n} be the values determined by this vertex labeling. Then

E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n))  (11)

is equal to n^{−k/2} if i_{j} < i_{j+1} whenever ϵ(j) = 1 and i_{j} > i_{j+1} whenever ϵ(j) = ∗, and is equal to zero otherwise. In terms of the orientations of the edges of the graph G as determined by ϵ(1), …, ϵ(k) in part (A) of Algorithm 2.5, the expectation (11) is n^{−k/2} if i_{j} < i_{j+1} whenever the arrow points to vertex v_{j} from vertex v_{j+1} and i_{j} > i_{j+1} whenever the arrow points from v_{j} to v_{j+1}, and the quantity (11) is zero otherwise. In terms of the orientations of the edges of the graph Q, the quantity (11) is n^{−k/2} if the arrows always point from the vertex labeled with the larger value to the vertex labeled with the smaller value. Thus for a fixed σ, the quantity

n^{−1} ∑_{distinct vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from {1,...,n}} E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n))  (12)

is equal to n^{−( \frac{k}{2} +1)} times the number of distinct vertex labelings of Q such that the label of vertex w_{i} of Q is less than the label of vertex w_{j} of Q whenever w_{i} ← w_{j}, where ← is the relation defined in part (C) of Algorithm 2.5. Because Q has \frac{k}{2} +1 vertices, as n → ∞ the quantity (12) converges to the volume with respect to Lebesgue measure of the set V of all (\frac{k}{2} + 1)–tuples (t_{1}, …, t_{\frac{k}{2} +1}) in the cube [0, 1]^{\frac{k}{2} +1} such that t_{i} < t_{j} whenever w_{i} ← w_{j}. Partitioning this cube into (\frac{k}{2} + 1)! wedges of equal measure corresponding to the different total orderings of the \frac{k}{2} + 1 co–ordinates, we see that V is the union of NTO(σ; ϵ(1), …, ϵ(k)) different wedges; hence (12) converges as n → ∞ to

\frac{1}{(\frac{k}{2} + 1)!} NTO(σ; ϵ(1), …, ϵ(k)).

Summing over all non–crossing pairings yields the expression (4) for the limit ∗– moment (3). □

Proof of Theorem 2.1. For any integer a, let

D_{n}(a) = \begin{cases}D_{n}^{a} & if a ≥ 0 \\ (D_{n}^{∗})^{−a} & if a < 0. \end{cases}

Then we must show that the limit

lim_{n→∞} τ_{n}(D_{n}(a_{1})T_{n}^{ϵ(1)} ⋯ D_{n}(a_{k})T_{n}^{ϵ(k)})  (13)

exists, for any k ∈ N, a_{1}, …, a_{k} ∈ Z and ϵ(1), …, ϵ(k) ∈ {∗, 1}. Writing t(i, j, ϵ; n) for the (i, j)th element of T_{n}^{ϵ} as in the proof of Lemma 2.4 and letting d(i; a, n) denote the ith diagonal entry of D_{n}(a), using the independence of D_{n} and T_{n} we have

τ_{n}(D_{n}(a_{1})T_{n}^{ϵ(1)} ⋯ D_{n}(a_{k})T_{n}^{ϵ(k)}) =
= n^{−1} ∑_{i_{1},...,i_{k}∈{1,...,n}} E(d(i_{1}; a_{1}, n) ⋯ d(i_{k}; a_{k}, n))·

· E(t(i_{1}, i_{2}, ϵ(1); n)t(i_{2}, i_{3}, ϵ(2); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n)) .

Since the hypotheses imply that

sup_{n∈N} sup_{i_{1},...,i_{k}∈{1,...,n}} |E(d(i_{1}; a_{1}, n) ⋯ d(i_{k}; a_{k}, n))| < ∞ ,

the same arguments as in the proof of Lemma 2.4 imply that the limit (13) is equal to the limit as n → ∞ of

n^{−1} ∑_{σ∈NCP(k) ; compatible} ∑_{distinct vertex labelings of Q(σ;ϵ(1),...,ϵ(k)) from \{1,...,n\}} \begin{matrix}( & E(d(i_{1}; a_{1}, n) ⋯ d(i_{k}; a_{k}, n))· \\ · E(t(i_{1}, i_{2}, ϵ(1); n) ⋯ t(i_{k}, i_{k+1}, ϵ(k); n))) , \end{matrix}  (14)

if the latter limit exists, where the first sum in (14) is over all non–crossing pairings σ of {1, …, k} that are compatible with ϵ(1), …, ϵ(k) and the second sum is over all labelings of the \frac{k}{2} + 1 vertices of Q = Q(σ; ϵ(1), …, ϵ(k)) with distinct elements of {1, …, n}, with each i_{j} being assigned the value of the label of the vertex of Q to which the vertex v_{j} of the k–gon graph G is mapped via the quotient mapping G → Q.

For fixed σ ∈ NCP(k) compatible with ϵ(1), …, ϵ(k), let w_{1}, …, w_{\frac{k}{2} +1} be the vertices of Q = Q(σ; ϵ(1), …, ϵ(k)) and for 1 ≤ p ≤ \frac{k}{2} + 1 let E(p) be the set of all j ∈ {1, …, k} such that the vertex v_{j} of G is mapped to w_{p} by the quotient map. Fix an arbitrary distinct vertex labeling of Q from {1, …, n} and let ι̂_{p} be the label of the vertex w_{p}. Because the entries of D_{n} are mutually independent, we have

E(d(i_{1}; a_{1}, n) ⋯ d(i_{k}; a_{k}, n)) = ∏_{p=1}^{\frac{k}{2} +1} E(∏_{j∈E(p)} d(ι̂_{p}; a_{j}, n)) = ∏_{p=1}^{\frac{k}{2} +1} ∫ λ^{r(p)}\overline{λ}^{s(p)}dμ(λ) ,

(15)

where

r(p) = ∑_{j∈E(p)} max(a_{j}, 0) s(p) = ∑_{j∈E(p)} max(−a_{j}, 0) .

The quantity (15) is thus independent of the particular distinct vertex labeling, and we will denote it E(σ; a_{1}, …, a_{k}; ϵ(i), … ϵ(k)). Using the same analysis as in the proof of Lemma 2.4, we see that the quantity (14) converges as n → ∞ to the complex number

\frac{1}{(\frac{k}{2} + 1)!} ∑_{σ∈NCP(k) ; compatible} E(σ; a_{1}, …, a_{k}; ϵ(i), … ϵ(k)) NTO(σ; ϵ(1), …, ϵ(k)) .  (16)

In particular, the limit (13) exists. □

In the case of ∗–moments of a DT–element, we easily obtain the following corollary.

Corollary 2.10. For every ℓ ∈ N and ϵ(1), …, ϵ(ℓ) ∈ {∗, 1} there is a polynomial P = P_{μ;ϵ(1),...,ϵ(ℓ)} in 1 + (ℓ + 1)^{2} variables and having nonnegative real coefficients such that if z is a DT(μ, c) element in a ∗–noncommutative probability space (𝔄, τ) then

τ(z^{ϵ(1)}z^{ϵ(2)} ⋯ z^{ϵ(ℓ)}) = P (c^{2}, (M_{μ}(r, s))_{r,s=0}^{ℓ}),

where M_{μ}(r, s) = ∫_{C} λ^{r}\overline{λ}^{s}dμ(λ). Furthermore, if we assign deg(M_{μ}(r, s)) = r + s and deg(c) = 1, then P is homogeneous of degree ℓ.

Corollary 2.11. Let μ and μ_{n} (n ∈ N) be compactly supported Borel probability measures on C such that μ_{n} converges in ∗–moments as n → ∞ to μ. Let c_{n} > 0 (n ∈ N) be such that lim_{n→∞} c_{n} = c ∈ (0, ∞). Let z_{n} be a DT(μ_{n}, c_{n}) element. Then z_{n} converges in ∗–moments as n → ∞ to a DT(μ, c) element.

Proposition 2.12. Let z be a DT(μ, c) element.

(i) If λ ∈ C\{0} then λz is DT(M_{λ}μ, |λ|c), where M_{λ}μ is the measure M_{λ}μ(B) = μ(λ^{−1}B).

(ii) z^{∗} is DT(\overline{μ}, c) where \overline{μ} is the measure \overline{μ}(B) = μ(\overline{B}).

Proof. Let Z_{n} = D_{n}+cT_{n} where D_{n} ∈ ℳ_{n} is a diagonal random matrix whose n diagonal entries are independent, each having distribution μ, where T_{n} ∈ UTGRM(n, \frac{1}{n}) and where D_{n} and T_{n} are independent. Then Z_{n} converges in ∗–moments to z.

For (i), λZ_{n} converges to λz in ∗–moments, as n → ∞. But each diagonal entry of λD_{n} has distribution M_{λ}μ and \frac{λ}{|λ|}T_{n} ∈ UTGRM(n, \frac{1}{n}), so λZ_{n} converges in ∗– moments to a DT(M_{λ}μ, |λ|c) element.

For (ii), Z_{n}^{∗} converges to z^{∗} in ∗–moments. Note that if U_{n} is the n×n permutation matrix effecting the permutation

1 ↦ n, 2 ↦ n − 1, …, n ↦ 1

then U_{n}Z_{n}^{∗}U_{n}^{∗} has the same distribution as \overline{D_{n}}+cT_{n}, hence Z_{n}^{∗} converges to a DT(\overline{μ}, c) element as n → ∞. □

Applying Theorem 3.6 of [6], we gain some important flexibility in choosing the diagonal part in random matrix approximations of DT–elements.

Theorem 2.13. Let μ be a compactly supported Borel probability measure on C. For every n ∈ N let D_{n} ∈ ℳ_{n} be a diagonal random matrix and denote by ν_{n} the joint distribution of the n diagonal entries of D_{n}. Let ν̃_{n} be the symmetrixation of ν_{n} and for p ∈ {1, …, n} let ν̃_{n}^{(p)} be the pth marginal distribution of ν̃_{n}. Suppose for every p ∈ N the measure ν̃_{n}^{(p)} converges in ∗–moments to ×_{1}^{p} μ.

For every n ∈ N let T_{n} ∈ UTGRM(n, \frac{1}{n}) be such that T_{n} and D_{n} are independent matrix–valued random variables. Let c > 0. Then D_{n} + cT_{n} converges in ∗–moments as n → ∞ to a DT(μ, c)–element.

Corollary 2.14. Let μ be a compactly supported Borel measure on C and let D_{n} ∈ M_{n}(C) be a sequence of non–random diagonal matrices with sup_{n} ∥D_{n}∥ < ∞. Assume that the ∗–moments of D_{n} converge to the the ∗–moments of μ as n → ∞. Let T_{n} ∈ UTGRM(n, \frac{1}{n}) and let c > 0. Then D_{n} + cT_{n} converges in ∗–moments to a DT(μ, c)–element.

Proof. We have D_{n} = diag(d_{1}^{(n)}, …, d_{n}^{(n)}) where d_{j}^{(n)} ∈ C. Let μ_{j}^{(n)} denote the dirac measure at d_{j}^{(n)}. Then with the notation of Theorem 2.13,

ν_{n} = μ_{1}^{(n)} × ⋯ × μ_{n}^{(n)} ν̃_{n} = \frac{1}{n!} ∑_{π∈S_{n}} μ_{π(1)}^{(n)} × ⋯ × μ_{π(n)}^{(n)} .

Hence the first marginal distribution is

ν̃_{n}^{(1)} = \frac{1}{n}(μ_{1}^{(n)} + ⋯ + μ_{n}^{(n)}) .

By the assumptions, ν̃_{n}^{(1)} converges in ∗–moments to μ as n → ∞. Since supp(ν̃_{n}^{(1)}) ⊆ K and supp(μ) ⊆ K, where K is the closed disk of radius R = sup_{n} ∥D_{n}∥ centered at the origin, by the Stone–Weierstrass theorem, we also have ν̃_{n}^{(1)} → μ in w^{∗}–topology as n → ∞.

We next consider the pth marginal distribution ν_{n}^{(p)} for p ≥ 2. If p = 2 we get

ν̃_{n}^{(2)} = \frac{1}{n(n − 1)} ∑_{i≠j} μ_{i}^{(n)} × μ_{j}^{(n)} ≤ \frac{1}{n(n − 1)} ∑_{i,j=1}^{n} μ_{i}^{(n)} × μ_{j}^{(n)} = \frac{n}{n − 1} ν̃_{n}^{(1)} × ν̃_{n}^{(1)} .

Hence ν̃_{n}^{(2)} − ν̃_{n}^{(1)} × ν̃_{n}^{(1)} = λ − ρ, where λ and ρ are the positive measures

λ = \frac{1}{n − 1}ν̃_{n}^{(1)} × ν̃_{n}^{(1)} ρ = \frac{n}{n − 1}ν̃_{n}^{(1)} × ν̃_{n}^{(1)} − ν̃_{n}^{(2)} ,

both of which have total mass 1/(n − 1). Thus the total variation ∥ν̃_{n}^{(2)} − ν̃_{n}^{(1)} × ν̃_{n}^{(1)}∥ of ν̃_{n}^{(2)} − ν̃_{n}^{(1)} × ν̃_{n}^{(1)} is at most 2/(n − 1) and

w^{∗}- lim_{n→∞} ν̃_{n}^{(2)} = w^{∗}- lim_{n→∞}(ν̃_{n}^{(1)} × ν̃_{n}^{(1)}) = μ × μ .

Since all the measures involved have support in K × K, convergence in ∗–moments of ν̃_{n}^{(2)} to μ × μ follows immediately.

The above argument can easily be generalized to p > 2. For n ≥ p we have

ν̃_{n}^{(p)} = \frac{(n − p)!}{n!} ∑ μ_{i_{1}}^{(n)} × ⋯ × μ_{i_{p}}^{(n)} ,

where the sum is over all p–tuples (i_{1}, …, i_{p}) of distinct integers in {1, …, n}. Hence

ν̃_{n}^{(p)} ≤ \frac{(n − p)! n^{p}}{n!} ×_{1}^{p} ν̃_{n}^{(1)} .

Arguing as above, we get

∥ν̃_{n}^{(p)} − ×_{1}^{p} ν̃_{n}^{(1)}∥ ≤ 2(\frac{(n − p)! n^{p}}{n!} − 1) → 0 as n → ∞,

from which we obtain convergence of ν̃_{n}^{(p)} to ×_{1}^{p} μ both in w^{∗}–topology and in ∗– moments. □

3. Examples of DT–elements

Recall that a circular element, as defined by Voiculescu (cf. [24]), is an element of a ∗–noncommutative probability space whose ∗–moment distribution is the same as x_{1} + ix_{2}, where x_{1} and x_{2} are free, centered semicircular elements having the same second moment. The circular free Poisson elements, one of which is a circular element, were defined in [6]. Specifically, the circular free Poisson element of parameter c ≥ 1 is an element of a ∗–noncommutative probability space having the same ∗– moment distribution as uh_{c} where u is a Haar unitary, h_{c} ≥ 0, h_{c}^{2} has the free Poisson distribution of parameter c and the pair u, h_{c} is ∗–free.

There it was shown, using a result of F. Dyson for the uppertriangularization of a nonsymmetric gaussian random matrix, that circular free Poisson elements are limits in ∗–moments of certain upper triangular random matrices, and that these random matrices satisfy properties implying the following theorem.

Theorem 3.1. Circular elements and circular free Poisson elements are DT–elements. More specifically, letting τ be the functional in the ∗–noncommutative probability space,

(i) a circular element z satisfying τ(z^{∗}z) = r^{2}, is DT(ρ_{r}, r) where ρ_{r} is uniform measure supported on the disk of radius r centered at 0;

(ii) a circular free Poisson element of parameter c is DT(ρ_{\sqrt{c−1},\sqrt{c}}, 1), where ρ_{\sqrt{c−1},\sqrt{c}} is uniform measure supported on the annulus having radii \sqrt{c − 1} and \sqrt{c}.

The elliptic elements form another class which includes the circular element. Their Brown measures were computed by F. Larsen in [15], (see also [4] and [11]) and their matrix models have been considered in [13]. We will say that an elliptic element is an element of a ∗–noncommutative probability space whose ∗–moment distribution is the same as ax_{1}+ibx_{2} for some a, b > 0, where x_{1} and x_{2} are free centered semicircular elements having second moment equal to 1. The remainder of this section is dedicated to showing that elliptic elements are DT–elements.

Theorem 3.2. An elliptic element ax_{1} + bx_{2} as above is a DT(ν_{a,b}, c_{a,b})–element, where ν_{a,b} is uniform measure supported on the solid ellipse

\{z ∈ C | \frac{(Re z)^{2}}{4a^{4}} + \frac{(Im z)^{2}}{4b^{4}} ≤ \frac{1}{a^{2} + b^{2}} \}

and where

c_{a,b} = \frac{2ab}{\sqrt{a^{2} + b^{2}}} .

The proof of this theorem will use the following result concerning random matrices, which is derived from Dyson’s result mentioned above.

Before stating this result, let us introduce some notation for subsets of M_{n}(C): • M_{n}(C)_{s.a.}, the self–adjoint matrices; • U_{n}, the unitary matrices;

• D_{n}, the diagonal matrices;

• T_{n}, the strictly upper triangular matrices, {(t_{ij})_{1≤i,j≤n} ∈ M_{n}(C) | t_{ij} =

0 if i ≥ j}. By Lebesgue measure on M_{n}(C), D_{n} and T_{n}, we will mean the product of Lebesgue measure on the real and imaginary parts of all their elements that are not a priori zero. By Lebesgue measure on M_{n}(C)_{s.a.}, we will mean of course the product of Lebesgue measure on the real diagonal entries and Lebesgue measure of the real and imaginary parts of all entries strictly above the diagonal.

An n × n random matrix H is said to belong to the class SGRM(n, σ^{2}) if it is self– adjoint, if its \frac{n(n−1)}{2} entries strictly above the diagonal are complex N(0, σ^{2}) random variables, if its n diagonal entries are real N(0, σ^{2}) random variables and if together they form a collection of \frac{n(n+1)}{2} of mutually independent random variables. On the other hand, an element of HURM(n) is an n × n random unitary matrix that is distributed according to Haar measure on the group of n × n unitaries.

Theorem 3.3. Let 0 < θ < \frac{π}{2}, let n ∈ N and let

Y_{θ}(n) = (cos θ)H_{1}(n) + i(sin θ)H_{2}(n),

where H_{1}(n) and H_{2}(n) are random matrices in the class SGRM(n, \frac{1}{n}) which are independent as matrix–valued random variables. Let

Z_{θ}(n) = U^{∗}(n)(D_{θ}(n) + sin(2θ)T (n))U(n) ,

where U(n) ∈ HURM(n), T (n) ∈ UTGRM(n, \frac{1}{n}) and D_{θ}(n) ∈ ℳ_{n} is a diagonal random matrix whose n diagonal entries have the distribution whose density with respect to Lebesgue measure on D_{n} is

ρ_{θ}(D) = K_{θ}( ∏_{1≤i<j≤n} |d_{i} − d_{j}|^{2}) exp ( − \frac{n}{2} ∑_{1≤i≤n} (\frac{(Re d_{i})^{2}}{cos^{2} θ} + \frac{(Im d_{i})^{2}}{sin^{2} θ} )),

(D = diag(d_{1}, …, d_{n}) ∈ D_{n})

(17)

for the appropriate constant K_{θ}. Then Y_{θ} and Z_{θ} have the same M_{n}(C)–valued distribution.

We would like to point out that the eigenvalue distribution (17) of Y_{θ}(n) was previously found by Hiai and Petz [13, Lemma 4.1.10] using different techniques.

Consider the action of U_{n} on M_{n}(C) by conjugation and let M_{n}(C)/U_{n} denote the measure space of equivalence classes. Every element of M_{n}(C)/U_{n} has a representative belonging to D_{n}+T_{n}, (and usually several of them). With respect to the quotient maps

\xymatrix{M_{n}(C) \ar[dr] &  & D_{n} + T_{n} \ar[dl] \\  & M_{n}(C)/U_{n}}

every measure μ on M_{n}(C) induces a measure on D_{n} + T_{n} having the same push– forward to M_{n}(C)/U_{n} as does μ.

Proof of Theorem 3.3. F. Dyson proved this (see [17, A.35]) when θ = π/4. For general θ, the distribution of Y_{θ}(n) has density with respect to Lebesgue measure on M_{n}(C)

ψ_{θ}(A + iB) = K′_{θ} exp ( − \frac{n}{2}Tr(\frac{A^{2}}{cos^{2} θ} + \frac{B^{2}}{sin^{2} θ})), (A, B ∈ M_{n}(C)_{s.a.}),

while the distribution of D_{θ}(n) + sin(2θ)T (n) has density with respect to Lebesgue measure on D_{n} + T_{n}

ϕ_{θ}(D + S) = K′′_{θ}ρ_{θ}(D) exp ( − \frac{n}{sin^{2}(2θ)}Tr(S^{∗}S)) (D ∈ D_{n}, S ∈ T_{n})

for appropriate constants K′_{θ} and K′′_{θ}. Since ψ_{θ} is invariant under unitary conjugation, as is the distribution of Z_{θ}(n), and since ϕ_{θ} agrees on elements of D_{n} + T_{n} that have the same image in M_{n}(C)/U_{n}, in order to prove the theorem it will suffice to show

∀X ∈ D_{n} + T_{n} \frac{ψ_{θ}(X)}{ψ\frac{π}{4} (X)} = c_{θ} \frac{ϕ_{θ}(X)}{ϕ\frac{π}{4} (X)} ,

where c_{θ} is a constant depending only on θ. Let X = D + S for D ∈ D_{n} and S ∈ T_{n}. Then X = A + iB for A, B ∈ M_{n}(C)_{s.a.} given by

A = Re (D) + \frac{1}{2}(S + S^{∗})
B = Im (D) + \frac{1}{2i}(S − S^{∗}) .

Since Re (D), S and S^{∗} are orthogonal in M_{n}(C) with respect to the inner product determined by the trace, we have

Tr(A^{2}) = Tr(Re (D)^{2}) + \frac{1}{2}Tr(S^{∗}S)

and similarly

Tr(B^{2}) = Tr(Im (D)^{2}) + \frac{1}{2}Tr(S^{∗}S) .

Hence

ψ_{θ}(D + S) = K′_{θ} exp ( − \frac{n}{2}Tr(\frac{Re (D)^{2}}{cos^{2} θ} + \frac{Im (D)^{2}}{sin^{2} θ} )) exp ( − \frac{n}{sin^{2}(2θ)}Tr(S^{∗}S)) .

From this one obtains that \frac{ψ_{θ}(D+S)}{ψ_{\frac{π}{4}} (D+S)} and \frac{ϕ_{θ}(D+S)}{ϕ_{\frac{π}{4}} (D+S)} coincide up to multiplication by a constant depending only on θ. □

We will need Hadamard’s Determinant Theorem:

Theorem 3.4 ([12]). Let A = (a_{ij})_{i,j=1}^{n} be a positive, semidefinite n × n matrix, i.e. A ∈ M_{n}(C)_{+} . Then

det(A) ≤ ∏_{i=1}^{n} a_{ii} .

Lemma 3.5. Let n ∈ N, n ≥ 2. Let dm(z) = d(Re z)d(Im z) denote the Lebesgue measure on C and let g : C → [0, ∞) be a Borel function such that

0 < ∫_{C} g(z)|z|^{k}dm(z) < ∞

for k = 0, 1, 2, . . .. Put

σ(z_{1}, …, z_{n}) = c ∏_{1≤i<j≤n} |z_{i} − z_{j}|^{2} ∏_{j=1}^{n} g(z_{j}) ,

where the normalization constant c is chosen so that σ is a probability density on C^{n}, i.e.

∫_{C^{n}} σ(z_{1}, …, z_{n})dm(z_{1}) ⋯ dm(z_{n}) = 1 .

For p ∈ {1, …, n}, let σ^{(p)} be the marginal density of the first p coordinates, i.e.

σ^{(p)}(u_{1}, …, u_{p}) = ∫_{C^{n−p}} σ(u_{1}, …, u_{p}, z_{p+1}, …, z_{n})dm(z_{p+1}) ⋯ dm(z_{n})

for p < n and σ^{(n)} = σ. Then

σ^{(p)}(u_{1}, …, u_{p}) ≤ \frac{(n − p)!}{n!}n^{p}σ^{(1)}(u_{1}) ⋯ σ^{(1)}(u_{p})

for all p ∈ {1, …, n} and u_{1}, …, u_{p} ∈ C.

Proof. The proof follows standard methods in random matrix theory, see [17, §5.2] for the real variable case and [17, §15.1] for the complex variable case with g(z) = e^{−|z|^{2}}. Let (P_{j}(z))_{j=0}^{∞} be the sequence of orthonormal polynomials obtained from the Gram– Schmidt orthonomalization process applied to 1, z, z^{2}, … in L^{2}(C, g dm), and put

ϕ_{j}(z) = \sqrt{g(z)}P_{j}(z) j = 0, 1, ….

Then (ϕ_{j}(z))_{j=0}^{∞} is an orthonormal sequence in L^{2}(C, dm). Using the Vandermonde determinant, one finds

σ(z_{1}, …, z_{n}) = \frac{1}{n!} | det ((ϕ_{i−1}(z_{j}))_{i,j=1}^{n})|^{2} = \frac{1}{n!} | ∑_{π∈S_{n}} sign(π) ∏_{j=1}^{n} ϕ_{π(j)−1}(z_{j})|^{2} ,

where S_{n} is the permitation group of \{1, …, n\}. The normalization constant \frac{1}{n!} can be determined from the fact that σ is a probability density. Now put

ψ(u, v) = ∑_{j=0}^{n−1} ϕ_{j}(u)\overline{ϕ_{j}(v)},  (u, v ∈ C) .  (18)

Then as in [17, pp. 80, 91-92], one finds

σ^{(p)}(u_{1}, …, u_{p}) = \frac{(n − p)!}{n!} det ((ψ(u_{i}, u_{j}))_{i,j=1}^{p}) .

In particular,

σ^{(1)}(u) = \frac{1}{n}ψ(u, u) .

By (18), (ψ(u_{i}, u_{j}))_{i,j=1}^{p} is a positive semidefinite matrix. Hence by Theorem 3.4,

σ^{(p)}(u_{1}, …, u_{p}) ≤ \frac{(n − p)!}{n!} ∏_{i=1}^{p} ψ(u_{i}, u_{i}) = \frac{(n − p)!}{n!}n^{p} ∏_{i=1}^{p} σ^{(1)}(u_{i}) .

□

Lemma 3.6. For each n ∈ N, let ν_{n} denote the probability measure on C^{n} with density given by (17) in Theorem 3.3, and for p ∈ {1, …, n} let ν_{n}^{(p)} be the joint marginal distribution of ν_{n} on the first p coordinates. Let μ be the uniform distribution on the solid ellipse

\{z ∈ C | \frac{(Re z)^{2}}{4 cos^{4} θ} + \frac{(Im z)^{2}}{4 sin^{4} θ} ≤ 1\} .

Then for every p ∈ N, ν_{n}^{(p)} converges to ×_{1}^{p} μ as n → ∞ both in ∗–moments and in

the w^{∗}–topology on Prob(C^{p}).

Proof. The case p = 1 is proved in [11]. Consider now the case p ≥ 2. By Lemma 3.5,

ν_{n}^{(p)} ≤ \frac{(n − p)!}{n!}n^{p}ν_{n}^{(1)} × ⋯ × ν_{n}^{(1)}, (n ≥ p) .

Hence, as in the proof of Corollary 2.14, we get

∥ν_{n}^{(p)} − ×_{1}^{p} ν_{n}^{(1)}∥ → 0 as n → ∞ ,

so in particular

lim_{n→∞} ν_{n}^{(p)} = lim_{n→∞} ×_{1}^{p} ν_{n}^{(1)} = ×_{1}^{p} μ

in the weak^{∗}–topology on Prob(C^{p}). However, ν_{n}^{(p)} is not compactly supported, so we must argue further to prove convergence in ∗–moments. Clearly, it suffices to consider convergence on real valued polynomials in z_{1}, …, z_{p}, \overline{z_{1}}, …, \overline{z_{p}}. Let h : C^{p} → R be such a polynomial. Choose C > 0 and d ∈ N such that

|h(z_{1}, …, z_{p})| ≤ C(1 + |z_{1}|^{2} + ⋯ + |z_{p}|^{2})^{d} .

Then

|h(z_{1}, …, z_{p})| ≤ C( ∑_{i=1}^{p}(1 + |z_{i}|^{2}))^{d} ≤ Cp^{d−1} ∑_{i=1}^{p}(1 + |z_{i}|^{2})^{d} ,

where the last inequality follows from the convexity of x ↦ x^{d} on [0, ∞). Put

g(z_{1}, …, z_{p}) = Cp^{d−1} ∑_{i=1}^{p}(1 + |z_{i}|^{2})^{d} .

Then g +h and g −h are non–negative, continuous functions of C^{p}. By appoximating g ± h from below with functions in C_{c}(C^{p})_{+}, it follows from the weak^{∗}–convergence

of ν_{n}^{(p)} to ×_{1}^{p} μ, that

∫_{C^{p}}(g ± h)d(×_{1}^{p} μ) ≤ lim inf_{n→∞} ∫_{C^{p}}(g ± h)dν_{n}^{(p)} .  (19)

Note that g is of the form g(z_{1}, …, z_{p}) = ∑_{i=1}^{p} f(z_{i}), where f : C → R is a polynomial in z and \overline{z}. Moreover, the one–dimensional marginal distributions of ν_{n}^{(p)} are all equal to ν_{n}^{(1)}. Using that ν_{n}^{(1)} converges to ν in ∗–moments, we therefore get

∫_{C^{p}} gd(×_{1}^{p} μ) = p ∫_{C} fdμ = lim_{n→∞} (p ∫_{C} fdν_{n}^{(1)}) = lim_{n→∞} ( ∫_{C} gdν_{n}^{(p)}) .

Hence by (19), we have

∫_{C^{p}} h d(×_{1}^{p} μ) ≤ lim inf_{n→∞} ∫_{C^{p}} h dν_{n}^{(p)} ≤ lim sup_{n→∞} ∫_{C^{p}} h dν_{n}^{(p)} ≤ ∫_{C^{p}} h d(×_{1}^{p} μ) .

This proves the convergence of ν_{n}^{(p)} to ×_{1}^{p} μ in ∗–moments. □

Proof of Theorem 3.2. In light of Proposition 2.12(i), we may without loss of generality assume a^{2} + b^{2} = 1, say a = cos θ and b = sin θ, some 0 < θ < \frac{π}{2}. By foundational results of Voiculescu [25], Y_{θ}(n) from Theorem 3.3 converges in ∗–moments as n → ∞ to the elliptic element (cos θ)x_{1}+i(sin θ)x_{2}. By Theorem 3.3, also D_{θ}(n)+sin(2θ)T (n) converges in ∗–moments as n → ∞ to this elliptic element. Let ν_{n} be the joint distribution of n complex variables having density (17) with respect to Lebesgue measure. In Lemma 3.6, it is shown that for every p ∈ N, the pth marginal distribution ν_{n}^{(p)} of ν_{n} converges in ∗–moments as n → ∞ to the p–fold Cartesian product of the uniform distribution on the solid ellipse

\{z ∈ C | \frac{(Re z)^{2}}{4 cos^{4} θ} + \frac{(Im z)^{2}}{4 sin^{4} θ} ≤ 1\}.

Consequently, the elliptic element ax_{1} + ibx_{2} is a DT(ν_{a,b}, sin(2θ))–element. □

4. DT–operators in finite von Neumann algebras

Our first task in this section is to show (Theorem 4.4) how every DT–element can be constructed in the von Neumann algebra L(F_{2}), (or more precisely, in the ∗– noncommutative probability space (L(F_{2}), τ), where τ is the tracial state on L(F_{2})), as D + cT where T is “the upper triangular part” of a semicircular element X and where D is from a “diagonal” algebra free from X. Then we prove a series of results culminating in what may be called our main technical theorem (Theorem 4.12), which shows that every DT–element can be realized as an upper triangular matrix having mutually free entries that are themselves circular elements or DT–elements. This theorem, as well as being pretty, is the main technical tool for proving decomposability of DT-operators in §5.

Let M be a von Neumann algebra with a normal faithful state τ, let λ : L^{∞}([0, 1]) → M be a normal, unital, injective ∗–homomorphism and let X ∈ M be a centered semicircular element with τ(X^{2}) = ν^{2} > 0 and such that X and the image of λ are free with respect to τ. (Thus the subalgebra of M generated by X and the image of λ is isomorphic to L(F_{2}), to which the restriction of τ is a trace.) For 0 ≤ a < b ≤ 1 let p[a, b] = λ(1_{[a,b]}) ∈ M.

Lemma 4.1. For ℓ ∈ N let

T_{ℓ} = ∑_{j=1}^{2^{ℓ}−1} p[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]Xp[\frac{j}{2^{ℓ}} , 1] .

Then T_{ℓ} converges in norm as ℓ → ∞ to an element T ∈ M, satisfying T + T^{∗} = X.

Proof. For example, T_{2} and T_{3} are represented by the shaded regions below.

T_{2}  T_{3}

We have

|T_{ℓ+1} − T_{ℓ}|^{2} = ∑_{j=1}^{2^{ℓ}} p[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ]Xp[\frac{2j−2}{2^{ℓ+1}}, \frac{2j−1}{2^{ℓ+1}}]Xp[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ] .

By results of Voiculescu [25],

p[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ]Xp[\frac{2j−2}{2^{ℓ+1}}, \frac{2j−1}{2^{ℓ+1}}]Xp[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ]

is the square of a circular element of norm 2^{(1−ℓ)/2}ν in p[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ]Mp[\frac{2j−1}{2^{ℓ+1}}, \frac{2j}{2^{ℓ+1}} ] with respect to the renormalization of τ. Thus ∥T_{ℓ+1} − T_{ℓ}∥ = 2^{(1−ℓ)/2}ν and T_{ℓ} converges in norm as ℓ → ∞.

In a similar manner, since

X − T_{ℓ} − T_{ℓ}^{∗} = ∑_{j=1}^{2^{ℓ}} p[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]Xp[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]

and p[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]Xp[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}] is a semicircular element of norm 2^{(2−ℓ)/2}ν, we find T +T^{∗} = X. □

Definition 4.2. The element T constructed in Lemma 4.1 from X and λ will be denoted T = 𝒰𝒯(X, λ).

Note that for any nonzero real number t, 𝒰𝒯(tX, λ) = |t|𝒰𝒯(X, λ).

For the remainder of this section, we will let X and λ be as described above, with the added convention that the second moment of X will be ν^{2} = 1.

Given n ∈ N and 0 ≤ a < b ≤ 1, let

P_{n}[a, b] = diag(\overbrace{\underbrace{0, …, 0}_{[na]}, 1, …, 1}^{[nb]}, 0, …, 0) ∈ M_{n}(C) .

Lemma 4.3. Let T = 𝒰𝒯(X, λ). Let D_{s} ∈ λ(L^{∞}[0, 1]) for all s in some set I. Suppose that D_{s}^{(k)} ∈ M_{2^{k}}(C) is a diagonal matrix (s ∈ I, k ∈ N), such that for all s ∈ I, ∥D_{s}^{(k)}∥ remains bounded as k → ∞ and such that the family

(D_{s}^{(k)})_{s∈I} , (P_{2^{k}}[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}}

converges in ∗–moments as k → ∞ to the family

(D_{s})_{s∈I} , (p[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} .

Let T^{(k)} ∈ UTGRM(2^{k}, 2^{−k}). Then the family

T^{(k)}, (D_{s}^{(k)})_{s∈I}  (20)

converges in ∗–moments as k → ∞ to the family

T, (D_{s})_{s∈I} .  (21)

Proof. We may assume the family (D_{s})_{s∈I} contains the identity and is closed under taking adjoints and under multiplication. Let X^{(k)} ∈ SGRM(2^{k}, 2^{−k}). By results of Voiculescu [25], the family

X^{(k)}, (D_{s}^{(k)})_{s∈I} , (P_{2^{k}}[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}}

converges in ∗–moments as k → ∞ to the family

X, (D_{s})_{s∈I} , (p[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} .

Therefore, letting

T_{ℓ}^{(k)} = ∑_{j=1}^{2^{ℓ}−1} P_{2^{k}}[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]X^{(k)}P_{2^{k}}[\frac{j}{2^{ℓ}} , 1] ,

the family

T_{ℓ}^{(k)}, (D_{s}^{(k)})_{s∈I}  (22)

converges in ∗–moments as k → ∞ to the family

T_{ℓ}, (D_{s})_{s∈I} .  (23)

Let δ_{ℓ}T^{(k)} = T^{(k)} − T_{ℓ}^{(k)} and for q ∈ {1, 2, 3, 4} let

A_{ℓ}^{(k)}(q) = \begin{cases}T_{ℓ}^{(k)} & if q = 1, \\ (T_{ℓ}^{(k)})^{∗} & if q = 2, \\ δ_{ℓ}T^{(k)} & if q = 3, \\ (δ_{ℓ}T^{(k)})^{∗} & if q = 4. \end{cases}

We will show that if m ∈ N, if q_{1}, …, q_{m} ∈ {1, 2, 3, 4} with q_{j} ∈ {3, 4} for at least one j ∈ {1, …, m} and if s_{1}, …, s_{m} ∈ I, then there is a constant C > 0, independent of k and ℓ, such that

|τ_{2^{k}}(A_{ℓ}^{(k)}(q_{1})D_{s_{1}}^{(k)} ⋯ A_{ℓ}^{(k)}(q_{m})D_{s_{m}}^{(k)})| ≤ C2^{−ℓ}  (24)

for all k ≥ ℓ ≥ 1. This estimate will finish the proof, because it will imply that for every ϵ(1), …, ϵ(m) ∈ {∗, 1}, there is a constant C′, independent of ℓ, such that

| lim_{k→∞} τ_{2^{k}}((T^{(k)})^{ϵ(1)}D_{s_{1}}^{(k)} ⋯ (T^{(k)})^{ϵ(m)}D_{s_{m}}^{(k)} − (T_{ℓ}^{(k)})^{ϵ(1)}D_{s_{1}}^{(k)} ⋯ (T_{ℓ}^{(k)})^{ϵ(m)}D_{s_{m}}^{(k)})| ≤ C′2^{−ℓ} .

Combined with the convergence of (22) to (23) and the norm convergence of T_{ℓ} to T as ℓ → ∞, this will imply convergence in ∗–moments of the family (20) to the family (21).

To prove the upper bound (24), we will make an analysis using graphs and pairings as did Voiculescu in [25] and we will use the fact that the proportion of nonzero entries in δ_{ℓ}T^{(k)} is on the order of 2^{−ℓ}. Fixing ℓ, we have

τ_{2^{k}}(A_{ℓ}^{(k)}(q_{1})D_{s_{1}}^{(k)} ⋯ A_{ℓ}^{(k)}(q_{m})D_{s_{m}}^{(k)})

= 2^{−k} ∑_{i_{1},...,i_{m}∈\{1,...,2^{k}\}} \begin{matrix}E(b^{(k)}(q_{1}; i_{0}, i_{1})d^{(k)}(s_{1}; i_{1}) & b^{(k)}(q_{2}; i_{1}, i_{2})d^{(k)}(s_{2}; i_{2}) ⋯ \\ ⋯ & b^{(k)}(q_{m}; i_{m−1}, i_{m})d^{(k)}(s_{m}; i_{m})) , \end{matrix}

where we use the convention i_{0} = i_{m}, where d^{(k)}(s; i) is the ith diagonal entry of D_{s}^{(k)} and where b(q; i, j) is the (i, j)th entry of A_{ℓ}^{(k)}(q). Let K = ∏_{j=1}^{m} sup_{k≥1} ∥D_{s_{j}}^{(k)}∥. Then

|E(b^{(k)}(q_{1}; i_{0}, i_{1})d^{(k)}(s_{1}; i_{1})b^{(k)}(q_{2}; i_{1}, i_{2})d^{(k)}(s_{2}; i_{2}) ⋯ b^{(k)}(q_{m}; i_{m−1}, i_{m})d^{(k)}(s_{m}; i_{m}))| ≤
≤ K|E(b^{(k)}(q_{1}; i_{0}, i_{1})b^{(k)}(q_{2}; i_{1}, i_{2}) ⋯ b^{(k)}(q_{m}; i_{m−1}, i_{m}))| .

By a generalized Hölder inequality,

|E(b^{(k)}(q_{1}; i_{0}, i_{1})b^{(k)}(q_{2}; i_{1}, i_{2}) ⋯ b^{(k)}(q_{m}; i_{m−1}, i_{m}))| ≤ 2^{−km/2} .  (25)

Moreover, the LHS of (25) is nonzero only if there is a pairing σ of {1, 2, …, m} such that {r, s} ∈ σ implies

(i) i_{r−1} = i_{s} and i_{r} = i_{s−1}, (ii) either {q_{r}, q_{s}} = {1, 2} or {q_{r}, q_{s}} = {3, 4}.

Therefore, an upper bound for the LHS of (24) is

K2^{−k(1+m/2)} ∑_{σ} N(σ) ,  (26)

where the sum is over all pairings σ satisfying (ii) for every {r, s} ∈ σ and where N(σ) is the number of choices of i_{1}, i_{2}, …, i_{m} ∈ {1, 2, …, 2^{k}} so that (i) holds for every {r, s} ∈ σ, (with i_{0} = i_{m}) and so that b^{(k)}(q_{j}; i_{j−1}, i_{j}) ≠ 0 for every j ∈ {1, …, m}. Choosing i_{1}, …, i_{m} such that (i) holds is equivalent to taking the quotient graph Q of the m–gon graph G according to σ as in Algorithm 2.5 part (B), (where of course k over there is m here), and assigning values from {1, …, 2^{k}} to the vertices of Q (thus assigning values to i_{1}, …, i_{m} according to the values assigned the images in Q of the corresponding vertices v_{1}, …, v_{m} in G). If σ is crossing, then from Lemma 2.9, Q has ≤ m/2 vertices and hence N(σ) ≤ 2^{km/2}. If σ is non–crossing, then Q has exactly 1 + m/2 vertices; however, examining Lemma 2.7, we find that in this case there is j with q_{j} ∈ {3, 4} and with i_{j−1} and i_{j} mapped to distinct vertices of Q. Since for a given value of i_{j} there are at most 2^{k−ℓ} values of i_{j−1} making b^{(k)}(q_{j}; i_{j−1}, i_{j}) nonzero, we find that N(σ) ≤ K2^{k(1+m/2)−ℓ}. Hence using (26), when k ≥ ℓ we get an upper bound of the form (24). □

Theorem 4.4. Let T = 𝒰𝒯(X, λ), (where X has second moment 1) and let c > 0. Take f ∈ L^{∞}[0, 1] and let D = λ(f). Then D + cT is a DT(μ, c)–element, where μ is the push–forward measure of Lebesgue measure by f.

Proof. One easily finds diagonal matrices D^{(k)} ∈ M_{2^{k}}(C) such that the family

D^{(k)}, (P_{2^{k}}[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}}

converges in ∗–moments as k → ∞ to the family

D, (p[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} .

Let T^{(k)} ∈ UTGRM(2^{k}, 2^{−k}). Then by Lemma 4.3, D^{(k)} + cT^{(k)} converges in ∗– moments to D + cT. On the other hand, by Corollary 2.14, D^{(k)} + cT^{(k)} converges in ∗–moments to a DT(μ, c)–element. □

Definition 4.5. A DT(μ, c)–operator is a DT(μ, c)–element in a W^{∗}–noncommutative probability space (M, ϕ) where ϕ is faithful. A DT–operator is an element that is a DT(μ, c)–operator for some μ and c.

Theorem 4.4 shows that for every pair (μ, c) there exist DT(μ, c)–operators.

Lemma 4.6. If T = 𝒰𝒯(X, λ) and 0 < t < 1 then

T p[0, t] = p[0, t]T p[0, t]  (27)
p[0, t]T p[t, 1] = p[0, t]Xp[t, 1] .  (28)

Proof. The identity (27) clearly holds when T is replaced by T_{ℓ}. Taking the limit as ℓ → ∞ proves (27). Then the identity (28) follows from (27) and X = T + T^{∗}. □

Lemma 4.7. Let T = 𝒰𝒯(X, λ) and let X′ = iT^{∗} − iT. Then X′ is a semicircular element of second moment 1, and X′ and the image of λ form a free pair. Furthermore,

−iT = 𝒰𝒯(X′, λ) .  (29)

Proof. Using Lemma 4.3 and taking, for example, the family (D_{s}^{(k)})_{s∈I} to be the family (P_{2^{k}}[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} itself, converging to the family

(D_{s})_{s∈I} = (p[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} ,

we find that the family

i(T^{(k)})^{∗} − iT^{(k)}, (P_{2^{k}}[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}}

converges in ∗–moments to

X′, (p[\frac{i}{2^{ℓ}} , \frac{j}{2^{ℓ}}])_{ℓ∈N, i,j∈\{0,1,...,2^{ℓ}\}} ,

where T^{(k)} ∈ UTGRM(2^{k}, 2^{−k}). By Voiculescu’s matrix model [25], it follows that X′ is a semicircular element of second moment 1 and X′ and the image of λ are free.

26 DT–operators

If 0 < t < 1, from Lemma 4.6 we get

p[0, t]X′p[t, 1] = p[0, t](−iT)p[t, 1] = −ip[0, t]Xp[t, 1] .

Hence

∑_{j=1}^{2^{ℓ}−1} p[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]X′p[\frac{j}{2^{ℓ}} , 1] = −iT_{ℓ} .

Letting ℓ → ∞ yields (29). □ Remark 4.8. What we have shown implies that if T = 𝒰𝒯(X, λ) then ∥T ∥ ≤ 2, because Re T = X/2 and Im T = X′/2 both have norm 1. In §8, we will show that actually ∥T ∥ = \sqrt{e}. Lemma 4.9. Let m ∈ N, 0 = s_{0} < s_{1} < ⋯ < s_{m} = 1 and

S = ∑_{j=1}^{m} p[s_{j−1}s_{j}]Xp[s_{j}, 1] .

Then with T = 𝒰𝒯(X, λ), we have ∥T − S∥ ≤ 2 max_{1≤j≤m}(s_{j} − s_{j−1})^{1/2}. Proof. We have

2 Re (T − S) = X − (S + S^{∗}) = ∑_{j=1}^{m} p[s_{j−1}, s_{j}]Xp[s_{j−1}, s_{j}] .

By results of Voiculescu [25], ∥p[s_{j−1}, s_{j}]Xp[s_{j−1}, s_{j}]∥ = 2(s_{j} − s_{j−1})^{1/2}. Hence

∥Re (T − S)∥ = max_{1≤j≤m}(s_{j} − s_{j−1})^{1/2} .

On the other hand, we have 2 Im (T − S) = X′ − (iS^{∗} − iS), where X′ = iT^{∗} − iT. Appealing to Lemmas 4.6 and 4.7,

iS = i ∑_{j=1}^{m} p[s_{j−1}, s_{j}]T p[s_{j}, 1] = ∑_{j=1}^{m} p[s_{j−1}, s_{j}]X′p[s_{j}, 1]

and ∥X′ − (iS^{∗} − iS)∥ = 2 max_{1≤j≤m}(s_{j} − s_{j−1})^{1/2}. □ Lemma 4.10. Let T = 𝒰𝒯(X, λ). Suppose B ⊆ [0, 1] is a Borel set of nonzero Lebesgue measure. Let α : [0, 1] → [0, 1] be defined by

α(t) = \frac{ρ([0, t) ∩ B)}{ρ(B)} ,

where ρ denotes Lebesgue measure on [0, 1]. Let q = λ(1_{B}) and let

λ̃ : L^{∞}[0, 1] → qλ(L^{∞}[0, 1]) ≅ L^{∞}(B)

be

λ̃(f) = qλ(f ∘ α) .

Then λ̃ is an injective, normal ∗–homomorphism and qT q = 𝒰𝒯(qXq, λ̃) .

Proof. The map α is monotone and continuous, hence measureable; therefore, λ̃ is a normal ∗–homomorphism. For any measureable set A ⊆ [0, 1], we have ρ(α^{−1}(A) ∩ B) = ρ(A)ρ(B). Hence if ρ(A) > 0, then λ̃(1_{A}) = λ(1_{α^{−1}(A)∩B}) ≠ 0 and λ̃ is injective. If 0 ≤ s < t ≤ 1, then up to measure zero, [s, t] ∩ B = α^{−1}([α(s), α(t)]) ∩ B, so

λ(1_{[s,t]})q = λ(1_{[α(s),α(t)]} ∘ α)q = λ̃(1_{[α(s),α(t)]}) .

Writing p[s, t] = λ(1_{[s,t]}) as usual and letting p̃[a, b] = λ̃(1_{[a,b]}), we thus have qp[s, t] = p̃[α(s), α(t)] and

qT_{ℓ}q = ∑_{j=1}^{2^{ℓ}−1} qp[\frac{j−1}{2^{ℓ}}, \frac{j}{2^{ℓ}}]Xp[\frac{j}{2^{ℓ}} , 1]q = ∑_{j=1}^{2^{ℓ}−1} p̃[α(\frac{j−1}{2^{ℓ}}), α(\frac{j}{2^{ℓ}})](qXq)p̃[α(\frac{j}{2^{ℓ}}), 1] .

By Voiculescu’s results [25], we have that, with respect to the trace τ(q)^{−1}τ↾_{qMq}, qXq is a semicircular element with second moment τ(q) and is free from the image of λ̃. Let T̃ = 𝒰𝒯(qXq, λ̃). Since

α(t) − α(s) ≤ \frac{t − s}{ρ(B)} , (0 ≤ s < t ≤ 1) ,

Lemma 4.9 gives ∥qT_{ℓ}q − T̃∥ ≤ 2^{1−(ℓ/2)}. Letting ℓ → ∞ yields T̃ = qT q. □

We will use the following freeness result concerning R–diagonal elements and Haar unitaries.

Proposition 4.11. Let (𝔅, ϕ) be a tracial W^{∗}–noncommutative probability space and let 1 ∈ A ⊆ 𝔅 be a unital subalgebra. Let N ∈ N, let b_{ij} (1 ≤ i < j ≤ N) be R– diagonal elements and let u_{i} (1 ≤ i ≤ N) be Haar unitaries such that

A, ({b_{ij}})_{1≤i<j≤N} , ({u_{i}})_{1≤i≤N}

is a ∗–free family. Then the family

(u_{i}^{∗}Au_{i})_{1≤i≤N} , ({u_{i}^{∗}b_{ij}u_{j}})_{1≤i<j≤N}

is ∗–free.

Proof. We may without loss of generality assume b_{ij} = v_{ij}|b_{ij}|, where each v_{ij} is a Haar unitary and the pair v_{ij}, |b_{ij}| is ∗–free. Then

u_{i}^{∗}b_{ij}u_{j} = (u_{i}^{∗}v_{ij}u_{j})(u_{j}^{∗}|b_{ij}|u_{j}) .

Since for a given j, the family

u_{j}^{∗}Au_{j} , ({u_{j}^{∗}|b_{ij}|u_{j}})_{1≤i<j}

is ∗–free, letting 𝔄 = W^{∗}(A ∪ {|b_{ij}| | 1 ≤ i < j ≤ N}), it will suffice to show the family

(u_{i}^{∗}𝔄u_{i})_{1≤i≤N} , ({u_{i}^{∗}v_{ij}u_{j}})_{1≤i<j≤N}

is ∗–free. We have that the family

𝔄, ({u_{i}})_{1≤i≤N} , ({v_{ij})_{1≤i<j≤N}

is ∗–free.

We will use the notation for any subalgebra C of 𝔅,

C^{o} = C ∩ ker ϕ.

Using freeness, we have

(u_{i}^{∗}𝔄u_{i})^{o} = u_{i}^{∗}𝔄^{o}u_{i}.

Given subsets X_{1}, …, X_{n} of 𝔅, we will use the notation

Λ^{o}(X_{1}, …, X_{n})  (30)

for the set of all words x_{1}x_{2} … x_{k} with k ≥ 1, x_{j} ∈ X_{i(j)} and i(1) ≠ i(2), i(2) ≠ i(3), …, i(k − 1) ≠ i(k). When refering to elements of (30), we will frequently and intentionally shift between formally defined words and the corresponding elements of 𝔅 obtained by performing the product operation.

Consider first Θ = Λ^{o}((u_{i}^{∗}𝔄^{o}u_{i})_{1≤i≤N}). Clearly Θ ⊆ Ψ, where Ψ is the set of all words belonging to Λ^{o}(𝔄^{o}, ({u_{i}, u_{i}^{∗}})_{1≤i≤N}) of length three or greater

• whose first letter is u_{i}^{∗}, some 1 ≤ i ≤ N,

• whose second letter is from 𝔄^{o},

• whose penultimate letter is from 𝔄^{o},

• whose last letter is u_{j}, some 1 ≤ j ≤ N. Thus it will suffice to show

Λ^{o}(Ψ, ({(u_{i}^{∗}v_{ij}u_{j})^{n} | n ∈ Z\{0}})_{1≤i<j≤N}) ⊆ ker ϕ .  (31)

However, taking a word x belonging to the LHS of (31), we see that some cancellation of neighboring letters of the form u_{j}u_{j}^{∗} may possibly be performed, however only in the following situations:

•  ⋯ 𝔄^{o}u_{j})(u_{j}^{∗}v_{jk} ⋯ (j < k)
•  ⋯ 𝔄^{o}u_{j})(u_{j}^{∗}v_{ij}^{∗} ⋯  (i < j)
•  ⋯ v_{ij}u_{j})(u_{j}^{∗}𝔄^{o} ⋯  (i < j)
• ⋯ v_{jk}^{∗}u_{j})(u_{j}^{∗}𝔄^{o} ⋯  (j < k)
•  ⋯ v_{ij}u_{j})(u_{j}^{∗}v_{jk} ⋯ (i < j < k)
•  ⋯ v_{ij}u_{j})(u_{j}^{∗}v_{pj}^{∗} ⋯ (i, p < j, i ≠ p)
• ⋯ v_{jk}^{∗}u_{j})(u_{j}^{∗}v_{jq} ⋯  (j < k, q, k ≠ q)
• ⋯ v_{jk}^{∗}u_{j})(u_{j}^{∗}v_{pj}^{∗} ⋯ (p < j < k).

Clearly, after making all such possible cancellations of u_{j}^{∗}u_{j} for all j, no further cancellations are possible, and we are left with an element of

Λ^{o}(𝔄^{o}, ({u_{i}, u_{i}^{∗}})_{1≤i≤N} , ({v_{ij}, v_{ij}^{∗}})_{1≤i<j≤N}) .

By freeness, this implies ϕ(x) = 0. □

Theorem 4.12. Let N ∈ N and let μ_{1}, …, μ_{N} be compactly supported Borel probability measures on C. Let c > 0 and suppose that in a W^{∗}–noncommutative probability space (M, τ),

(a_{k})_{k=1,}^{N} (b_{ij})_{1≤i<j≤N}

is a ∗–free family, where a_{k} is DT(μ_{k}, \frac{c}{\sqrt{N}}) and each b_{ij} is circular with τ(|b_{ij}|^{2}) = \frac{c^{2}}{N}. Let

Z = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1N} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{N−1,N} \\ 0 & ⋯ & 0 & a_{N} \end{matrix} ) ∈ M_{N}(M) .

Then with respect to the state τ ∘ tr_{N} on M_{N}(M), Z is a DT(μ, c)–element, where μ = \frac{1}{N} (μ_{1} + ⋯ + μ_{N}).

Proof. In light of Proposition 2.12, we may without loss of generality fix c = 1. Let (𝔄, τ) be a tracial W^{∗}–noncommutative probability space having semicircular elements x_{i} with τ(x_{i}^{2}) = 1, (1 ≤ i ≤ N), circular elements y_{ij} with τ(|y_{ij}|^{2}) = 1, (1 ≤ i < j ≤ N), and with a normal, injective ∗–homomorphism λ : L^{∞}[0, 1] → 𝔄 such that the family

({x_{i}})_{1≤i≤N} , ({y_{ij}, y_{ij}^{∗}})_{1≤i<j≤N} , λ(L^{∞}[0, 1])

is free. By random matrix results of Voiculescu [25], it follows that, with respect to the obvious tracial state on M_{N}(𝔄), the element

X = \frac{1}{\sqrt{N}} (\begin{matrix}x_{1} & y_{12} & ⋯ & y_{1N} \\ y_{12}^{∗} & x_{2} & ... & ⋮ \\ ⋮ & ... & ... & y_{N−1,N} \\ y_{1N}^{∗} & ⋯ & y_{N−1,N}^{∗} & a_{N} \end{matrix} ) ∈ M_{N}(𝔄)

is semicircular with second moment 1 and is free from the image of κ, where κ : L^{∞}[0, 1] → M_{N}(𝔄) is given by

κ(f) = diag(λ(f · 1_{[0,1/N]}), λ(f · 1_{[1/N,2/N]}), …, λ(f · 1_{[(N−1)/N,1]})) .

Let f_{i} ∈ L^{∞}[0, 1] be such that the push–forward of Lebesgue measure under f_{i} is μ_{i}, Let d_{i} = λ(f_{i}) and let D = diag(d_{1}, …, d_{N}) ∈ M_{N}(𝔄). Then D = κ(f) for some f ∈ L^{∞}[0, 1] where the push forward of Lebesgue measure under f is μ. Let T = 𝒰𝒯(X, κ). By Theorem 4.4, D + T is a DT(μ, 1)–element. From Lemma 4.6,

D + T = ∑_{1≤i≤j≤N} e_{ii}(D + T)e_{jj} = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1N} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{N−1,N} \\ 0 & ⋯ & 0 & a_{N} \end{matrix} ) ,

where b_{ij} = \frac{1}{\sqrt{N}} y_{ij} and a_{i} = d_{i} + e_{ii}T e_{ii}. By Lemma 4.10,

e_{ii}T e_{ii} = 𝒰𝒯(e_{ii}Xe_{ii}, λ) = 𝒰𝒯(\frac{1}{\sqrt{N}} x_{i}, λ) = \frac{1}{\sqrt{N}} 𝒰𝒯(x_{i}, λ) ,

so by Theorem 4.4 again, a_{i} is a DT(μ_{i}, \frac{1}{\sqrt{N}})–element.

Clearly, the family

{a_{i} | 1 ≤ i ≤ N}, ({b_{ij}})_{1≤i<j≤N}

is ∗–free, however the various a_{i} are not free among themselves. We may without loss of generality assume there are Haar unitaries u_{1}, …, u_{N} ∈ 𝔄 so that

({u_{i}})_{1≤i≤N} {a_{i} | 1 ≤ i ≤ N}, ({b_{ij}})_{1≤i<j≤N}  (32)

is a ∗–free family. Let U = diag(u_{1}, …, u_{N}) ∈ M_{N}(𝔄). Then U^{∗}(D + T)U is a DT(μ, 1)–element and

U^{∗}(D + T)U = (\begin{matrix}ã_{1} & b̃_{12} & ⋯ & b̃_{1N} \\ 0 & ã_{2} & ... & ⋮ \\ ⋮ & ... & ... & b̃_{N−1,N} \\ 0 & ⋯ & 0 & ã_{N} \end{matrix} ) ,

where ã_{i} = u_{i}^{∗}a_{i}u_{i} and b̃_{ij} = u_{i}^{∗}b_{ij}u_{j}^{∗}. By Proposition 4.11, the family

{ã_{i} | 1 ≤ i ≤ N}, ({b̃_{ij}})_{1≤i<j≤N}

is ∗–free. We have therefore realized a DT(μ, 1)–element as a matrix having ∗–free entries of the desired form. □

5. Decomposability of DT–operators

In this section, we apply the results of §4 to show that every DT–operator is strongly decomposable.

Lemma 5.1. If A and B are bounded operators on a Hilbert space and if A is normal then

σ(A + B) ⊆ {z ∈ C | d(z, σ(A)) ≤ ∥B∥} .

Proof. If λ ∈ C and d(λ, σ(A)) > ∥B∥ then ∥(A − λ)^{−1}∥ < ∥B∥^{−1}. Therefore (A − λ)^{−1}(A + B − λ) = 1 − (A − λ)^{−1}B is invertible, and λ ∉ σ(A + B). □

Theorem 5.2. Let μ be a compactly supported Borel probability measure on C and let c > 0. If a is a DT(μ, c) operator then σ(a) = supp(μ).

Proof. Let (M, τ) be a W^{∗}–noncommutative probability space, with τ a faithful trace. The spectrum of an operator a ∈ M depends only on its ∗–moment distribution with respect to τ. Let N ∈ N and let (a_{k})_{k=1}^{N}, (b_{ij})_{1≤i,j≤N} be a ∗–free family of elements of M, where each a_{k} is DT(μ, \frac{c}{\sqrt{N}}) and each b_{ij} is circular with τ(|b_{ij}|^{2}) = \frac{c^{2}}{N}. Let

x = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1N} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{N−1,N} \\ 0 & ⋯ & 0 & a_{N} \end{matrix} ) ∈ M_{N}(M) .

Then by Theorem 4.12, x is DT(μ, c) with respect to τ ∘ tr_{N}. Since M is a finite von Neumann algebra, we have σ(x) = ⋃_{k=1}^{N} σ(a_{k}) = σ(a_{1}). We have shown that the spectrum of a DT(μ, c) operator is the same as that of a DT(μ, \frac{c}{\sqrt{N}}) operator.

Let us use Theorem 4.4, to realize a DT(μ, c)–operator as D + cT, for D a normal operator with spectrum supp(μ) and as noted in Remark 4.8, ∥T ∥ ≤ 2. We have σ(D + cT) = σ(D + \frac{c}{\sqrt{N}} T) for all N ∈ N. Using Lemma 5.1 we thus get

σ(D + cT) ⊆ \{z ∈ C | d(z, supp(μ)) ≤ \frac{2c}{\sqrt{N}} \}

for all N ∈ N. Hence σ(D + cT) ⊆ supp(μ).

On the other hand, suppose for contradiction that λ ∈ supp(μ)\σ(D + cT). Then there is ϵ > 0 such that the ball B_{ϵ}(λ) of radius ϵ around λ is disjoint from σ(D+cT). Let N ∈ N be such that N ≥ μ(B_{ϵ}(λ))^{−1}. Using Theorem 4.12, we may take (M, τ) = (M_{N}(N), τ_{N} ∘ tr_{N}) and

D + cT = x = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1N} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{N−1,N} \\ 0 & ⋯ & 0 & a_{N} \end{matrix} ) ∈ M_{N}(N) ,

where N is a von Neumann algebra with faithful normal tracial state τ_{N} and where a_{1} is DT(μ_{1}, \frac{c}{\sqrt{N}}) with supp(μ_{1}) ⊆ B_{ϵ}(λ). But σ(D + cT) = ⋃_{k=1}^{N} σ(a_{k}) and σ(a_{1}) ⊆ supp(μ_{1}). Since σ(a_{1}) is nonempty, this implies that B_{ϵ}(λ) meets the spectrum of D + cT, which is a contradiction. □

The next result relates upper triangular decompositions to local spectral subspaces.

Proposition 5.3. Let T be a bounded operator on a separable Hilbert space ℋ = ℋ_{1} ⊕ ℋ_{2} and suppose ℋ_{1} ⊕ 0 is T –invariant, so that T = ( \begin{matrix}T_{11} & T_{12} \\ 0 & T_{22} \end{matrix} ) where T_{ij} : ℋ_{j} → ℋ_{i}. Let A ⊆ C. If σ(T_{11}) ⊆ A, then ℋ_{1} ⊕ 0 ⊆ ℋ_{T} (A). On the other hand, if σ(T_{22}) ∩ A = ∅, then ℋ_{T} (A) ⊆ ℋ_{1} ⊕ 0.

Proof. Write elements ξ of ℋ as ξ = ( \begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix} ) with ξ_{i} ∈ ℋ_{i}. Suppose σ(T_{11}) ⊆ A. For

ξ_{1} ∈ ℋ_{1}, taking f(λ) = ( \begin{matrix}(T_{11}−λ)^{−1}ξ_{1} \\ 0 \end{matrix} ) we have (T − λ)f(λ) = ( \begin{matrix}ξ_{1} \\ 0 \end{matrix} ), (λ ∈ C\σ(T_{11})).

Thus σ_{T} (( \begin{matrix}ξ_{1} \\ 0 \end{matrix} )) ⊆ σ(T_{11}) ⊆ A and therefore ( \begin{matrix}ξ_{1} \\ 0 \end{matrix} ) ∈ ℋ_{T} (A).

Now suppose σ(T_{22}) ∩ A = ∅ and ξ = ( \begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix} ) ∈ ℋ_{T} (A). Let f(λ) = ( \begin{matrix}f_{1}(λ) \\ f_{2}(λ) \end{matrix} ) be

analytic such that (T − λ)f(λ) = ξ, (λ ∈ C\σ_{T} (ξ)). However, as σ_{T} (ξ) ∩ σ(T_{22}) = ∅ and as T has the single–valued extension property, (since ℋ is separable), the analytic function f_{2} extends to an entire function f̃_{2} : C → ℋ_{2} such that (T_{22} − λ)f_{2}(λ) = ξ_{2} for all λ ∈ C. Since f_{2}(λ) = (T_{22} − λ)^{−1}ξ_{2} → 0 as |λ| → ∞, by Liouville’s theorem f_{2} is the zero function and ξ_{2} = 0; thus ( \begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix} ) ∈ ℋ_{1} ⊕ 0. □

For the rest of this section, Z will be a DT(μ, c)–operator in a W^{∗}–noncommutative probability space (M, τ), with M ⊆ B(ℋ), ℋ separable and τ faithful.

Theorem 5.4. Let B be a Borel subset of C and let p = p_{Z}(B) be the projection from ℋ onto \overline{ℋ_{Z}(B)}. Then

(i) p ∈ M, Zp = pZp and τ(p) = μ(B)

(ii) if μ(B) > 0 then Zp is DT(μ(B)^{−1}μ↾_{B}, c\sqrt{μ(B)}) with respect to μ(B)^{−1}τ↾_{pMp} (iii) if μ(B) < 1 then (1 − p)Z is DT(μ(B^{c})^{−1}μ↾_{B^{c}}, c\sqrt{μ(B^{c})}) with respect to

μ(B^{c})^{−1}τ↾_{(1−p)M(1−p)} .

Proof. ℋ_{Z}(B) is Z–hyperinvariant (see [16, 1.2.16(a)]) and therefore so is \overline{ℋ_{Z}(B)}; this implies p ∈ M and Zp = pZp. Let n ∈ N and let k ∈ {0, 1, …, n − 1}, let K_{n} ⊆ C be compact and let U_{n} ⊆ C be open such that K_{n} ⊆ B ⊆ U_{n} and

\frac{k}{n} ≤ μ(K_{n}) ≤ μ(B) ≤ μ(U_{n}) ≤ \frac{k + 1}{n} .

Consider the probability measures

ν_{n} = μ(K_{n})^{−1}μ↾_{K_{n}} if μ(K_{n}) > 0 ,
ν′_{n} = μ(U_{n}^{c})^{−1}μ↾_{U_{n}^{c}} if μ(U_{n}) < 1 ,
ν′′_{n} = nμ↾_{U_{n}\K_{n}} + (nμ(K_{n}) − k)ν_{n} + (k + 1 − nμ(U_{n}))ν′_{n} .

Then μ = \frac{1}{n}(kν_{n} + ν′′_{n} + (n − k − 1)ν′_{n}) and hence by Theorem 4.12 there is a DT(μ, c) operator

Z_{n} = (\begin{matrix}a_{1} & b_{12} & ⋯ & b_{1n} \\ 0 & a_{2} & ... & ⋮ \\ ⋮ & ... & ... & b_{n−1,n} \\ 0 & ⋯ & 0 & a_{n} \end{matrix})

in (M_{n}(N), τ_{N} ∘ tr_{n}) where in (N , τ_{N}), a_{i} is DT(ν_{n}, \frac{c}{\sqrt{n}}) if i ≤ k and is DT(ν′_{n}, \frac{c}{\sqrt{n}}) if i ≥ k + 2, a_{k+1} is DT(ν′′_{n}, \frac{c}{\sqrt{n}}), b_{ij} is circular with τ_{N} (|b_{ij}|^{2}) = \frac{c^{2}}{n} and the family (a_{i})_{i=1}^{n}, (b_{ij})_{1≤i<j≤n} is ∗–free.

Let

q_{n} = diag(\underbrace{1, …, 1}_{k}, \underbrace{0, …, 0}_{n−k})
q′_{n} = diag(\underbrace{1, …, 1}_{k+1}, \underbrace{0, …, 0}_{n−k−1})

in M_{n}(N). Applying Theorem 4.12 and Theorem 5.2, if k > 0 then Z_{n}q_{n} is DT(ν_{n}, c\sqrt{\frac{k}{n}}) in (M_{k}(N), τ_{N} ∘ tr_{k}) and σ(Z_{n}q_{n}) ⊆ K_{n}. Similarly, if k < n − 1 then (1 − q′_{n})Z_{n} is

DT(ν′_{n}, c\sqrt{\frac{n−k−1}{n}}) in (M_{n−k−1}(N), τ_{N} ∘ tr_{n−k−1}) and σ((1 − q′_{n})Z_{n}) ⊆ U_{n}^{c}. But Proposition 5.3 then gives q_{n} ≤ p_{Z_{n}}(B) ≤ q′_{n}. Letting n → ∞, this shows that τ(p) = μ(B). Thus (i) is proved.

If μ(B) > 0 then Z_{n}q_{n} converges in ∗–moments to Zp as n → ∞. Indeed, for ϵ(1), …, ϵ(ℓ) ∈ {∗, 1} we have

τ(pZ^{ϵ(1)} ⋯ pZ^{ϵ(ℓ)}p) = τ_{N} ∘ tr_{N}(p_{Z_{n}}(B)Z_{n}^{ϵ(1)} ⋯ p_{Z_{n}}(B)Z_{n}^{ϵ(ℓ)}p_{Z_{n}}(B))

and this quantity differs from τ_{N} ∘ tr_{N}(q_{n}Z_{n}^{ϵ(1)} ⋯ q_{n}Z_{n}^{ϵ(ℓ)}q_{n}) by an amount no greater than ∥q_{n} − p_{Z_{n}}(B)∥_{2}Q_{ℓ}(∥Z_{n}∥), where Q_{ℓ} is a polynomial independent of n and with positive coefficients. Since ∥q_{n} − p_{Z_{n}}(B)∥_{2} ≤ n^{−1/2} → 0 as n → ∞, this shows that Z_{n}q_{n} converges to Zp. Clearly, ν_{n} converges in ∗–moments as n → ∞ to μ(B)^{−1}μ↾_{B}.

Now an application of Corollary 2.11 shows that Zp is DT(μ(B)^{−1}μ↾_{B}, c\sqrt{μ(B)}), and (ii) is proved.

For (iii), one shows similarly that if μ(B) < 1 then (1 − q′_{n})Z_{n} converges in ∗– moments to (1 − p)Z as n → ∞, ν′_{n} converges in ∗–moments to μ(B^{c})^{−1}μ↾_{B^{c}} and thus that (1 − p)Z is DT(μ(B^{c})^{−1}μ↾_{B^{c}}, c\sqrt{μ(B^{c})}). □

L.G. Brown [5] discovered a measure supported inside the spectrum of an arbitrary element of a finite von Neumann algebra, which in the finite dimensional case reduces to the eigenvalue distribution weighted according to generalized multiplicity of eigenvalues.

Corollary 5.5. The Brown measure of a DT(μ, c) operator is μ.

Proof. Let Z be a DT(μ, c)–operator, and let ν_{Z} be the Brown measure of Z. From the theorem just proved and [5, Theorem 4.3], it follows that ν_{Z}(F) ≥ τ(p_{Z}(F)) = μ(F) for every closed subset F of C. Thus ν_{Z} = μ. □

The next result is a converse to Theorem 5.4, which we will use in §6.

Theorem 5.6. (i) Let F ⊆ C be a closed subset and suppose q ∈ M is a projec-

tion satisfying Zq = qZq, τ(q) = μ(F) and σ(Z↾_{qℋ} ) ⊆ F. Then q = p_{Z}(F). (ii) Let B ⊆ C be a Borel subset and suppose q ∈ M is a projection satisfying

Zq = qZq, τ(q) = μ(B) and that Zq is a DT(μ(B)^{−1}μ↾_{B}, c′)–operator with respect to μ(B)^{−1}τ↾_{qMq} for some c′ > 0. Then q = p_{Z}(B) and c′ = c\sqrt{μ(B)}.

Proof. In (i), Proposition 5.3 implies q ≤ p_{Z}(F). But τ(q) = τ(p_{Z}(F)), so q = p_{Z}(F). By the definition of spectral subspaces we have

ℋ_{Z}(B) = ⋃_{F} ℋ_{Z}(F),  (33)

where the union is over all closed subsets F of C such that F ⊆ B. Regard Zq as an operator on qℋ and let F ⊆ B be a closed subset of C. Then by Theorem 5.4, the Zq–invariant projection p_{Zq}(F) satisfies τ(p_{Zq}(F)) = μ(F) and σ(Z↾_{p_{Zq}(F)ℋ} ) = F. But p_{Zq}(F) is also Z–invariant, and hence part (i) implies p_{Z}(F) = p_{Zq}(F) ≤ q. Using (33) we get p_{Z}(B) ≤ q. Now τ(p_{Z}(B)) = τ(q) implies p_{Z}(B) = q. □

The following result shows that Z has property (C) of Dunford; (see [16, 1.2.18]).

Lemma 5.7. If F is a closed subset of C then ℋ_{Z}(F) is closed.

Proof. If μ(F) = 0 then by Theorem 5.4, \overline{ℋ_{Z}(F)} = ℋ_{Z}(F) = \{0\}, so suppose μ(F) > 0. We must show \overline{ℋ_{Z}(F)} = ℋ_{Z}(F) or, in other words, that if ξ ∈ \overline{ℋ_{Z}(F)} then there is an analytic function f : C\F → ℋ such that (Z − λ)f(λ) = ξ, (λ ∈ C\F). Letting p = p_{Z}(F) be the projection from ℋ onto \overline{ℋ_{Z}(F)}, then Zp = pZp and by Theorem 5.4 and Theorem 5.2, σ(Zp) ⊆ F, where the spectrum is for Zp as an element of pMp. Now given ξ ∈ pℋ let f(λ) = (pZp − λp)^{−1}ξ (λ ∈ C\F), where the inverse is taken in pMp. Then f is analytic and (Z − λ)f(λ) = ξ. □

Theorem 5.8. Z is strongly decomposable.

Proof. We will first show that Z is decomposable, and we will use the characterization of decomposability found at [16, 1.2.23(b)], which is essentially due to Jafarian and Vasilescu [14]. Since we already showed in Lemma 5.7 that Z has property (C), it remains only to show that for every closed subset F of C, the spectrum of the operator Z/ℋ_{Z}(F) induced by Z on the quotient space ℋ/ℋ_{Z}(F) is contained in \overline{σ(Z)\F}. Note that σ(Z) = supp(μ). Consider first the trivial cases μ(F) ∈ \{0, 1\}. From Theorem 5.4 we have ℋ_{Z}(F) = {0} if and only if μ(F) = 0, and in this case int(F) = ∅ so \overline{σ(Z)\F} = σ(Z), while σ(Z/ℋ_{Z}(F)) = σ(Z). If μ(F) = 1 then ℋ_{Z}(F) = ℋ and Z/ℋ_{Z}(F) is the operator on the Hilbert space {0}, which has empty spectrum.

Suppose 0 < μ(F) < 1 and let p = p_{Z}(F) be the projection from ℋ onto ℋ_{Z}(F). Then ℋ/ℋ_{Z}(F) is canonically isomorphic to (1 − p)ℋ in such a way that Z/ℋ_{Z}(F) corresponds to (1 − p)Z ∈ B((1 − p)ℋ). By Theorems 5.4 and 5.2, the spectrum of (1 − p)Z is supp(μ↾_{F^{c}}) = \overline{supp(μ)\F} = \overline{σ(Z)\F}. Thus Z is decomposable.

In order to show that Z is strongly decomposable we must show that for every closed subset F of C, the restriction Z↾_{ℋ_{Z}(F)} of Z to ℋ_{Z}(F) is decomposable. However, by Theorem 5.4, either ℋ_{Z}(F) = {0} or Z↾_{ℋ_{Z}(F)} is itself a DT–operator. In either case, Z↾_{ℋ_{Z}(F)} is decomposable. □

6. Von Neumann algebras generated by DT–operators

Theorem 4.4 allows us realize a DT(μ, c)–operator as an element D + cT in the II_{1}–factor L(F_{2}), where we have some flexibility in choosing D. It is natural to ask: what sort of II_{1}–factors do these DT–operators generate? It is this question that is addressed in this section. We first consider the von Neumann algebra, which we will denote Q, generated by T = 𝒰𝒯(X, λ) inside L(F_{2}) (cf Definition 4.2). We show (Theorem 6.1) that Q is an irreducible subfactor of L(F_{2}), but we are presently unable to decide whether Q is all of L(F_{2}). This question is closely bound up with the question of whether T has nontrivial hyperinvariant subpaces.

In a tracial W^{∗}–noncommutative probability space (M, τ), let X be a semicircular operator and let λ : L^{∞}[0, 1] → M be a normal, unital, injective ∗–homomorphism whose image is free from X. Thus X and the image of λ together generate a copy of L(F_{2}).

Theorem 6.1. Let T = 𝒰𝒯(X, λ) be the operator as constructed in Lemma 4.1 inside the von Neumann algebra factor N = ({X} ∪ λ(L^{∞}[0, 1]))′′ ≅ L(F_{2}). Then the von Neumann algebra generated by T is an irreducible subfactor of N, and is in particular a factor.

Proof. We use the convention, given a state ϕ on a von Neumann algebra M, that a ↦ â denotes the defining mapping of M into L^{2}(M, ϕ).

The W^{∗}–subalgebra of N generated by X is canonically isomorphic to L^{∞}[−2, 2]. Since W^{∗}(X) and the image of λ together freely generate N, the free product construction [23] identifies L^{2}(N , τ↾_{N}) with the Hilbert space

ℋ = C ⊕ ⨁_{\begin{matrix}n≥1 \\ ι_{1},...,ι_{n}∈\{1,2\} \\ ι_{j}≠ι_{j+1} \end{matrix}} ℋ_{ι_{1}}^{o} ⊗ ⋯ ⊗ ℋ_{ι_{n}}^{o} ,

where

ℋ_{1}^{o} = L^{2}[0, 1] ⊖ Cˆ1 = (λ(L^{∞}[0, 1]) ∩ ker τ)
ℋ_{2}^{o} = L^{2}[−2, 2] ⊖ Cˆ1 = (W^{∗}(X) ∩ ker τ) .

For example, in this identification, an element (a_{1}b_{1}a_{2}b_{2} ⋯ a_{n}b_{n})ˆ of L^{2}(N , τ↾_{N}), where a_{i} ∈ W^{∗}(X) ∩ ker τ and b_{i} ∈ λ(L^{∞}[0, 1]) ∩ ker τ, corresponds to the element â_{1} ⊗ b̂_{1} ⊗ ⋯ ⊗ â_{n} ⊗ b̂_{n} of the direct summand ℋ_{2}^{o} ⊗ ℋ_{1}^{o} ⊗ ⋯ ⊗ ℋ_{2}^{o} ⊗ ℋ_{1}^{o} of ℋ. The map L^{∞}[0, 1] ⊙ L^{∞}[0, 1] → N, where ⊙ denotes the algebraic tensor product, defined by f ⊗ g ↦ λ(f)Xλ(g), gives rise to an isometry W : L^{2}[0, 1] ⊗ L^{2}[0, 1] → ℋ at the L^{2}–level. Specifically,

W (v_{1} ⊗ v_{2}) =⟨v_{1}, ˆ1⟩X̂⟨v_{2}, ˆ1⟩ + (v_{1} − ⟨v_{1}, ˆ1⟩ˆ1) ⊗ X̂⟨v_{2}, ˆ1⟩

+ ⟨v_{1}, ˆ1⟩X̂ ⊗ (v_{2} − ⟨v_{2}, ˆ1⟩ˆ1) + (v_{1} − ⟨v_{1}, ˆ1⟩ˆ1) ⊗ X̂ ⊗ (v_{2} − ⟨v_{2}, ˆ1⟩ˆ1).

In other words, W (v_{1}⊗v_{2}) = v_{1}⊗X̂ ⊗v_{2} if we make the convention that ˆ1 is absorbed by tensor products. Then from the definition of T, we have T̂ = W (h), where h ∈ L^{2}[0, 1] ⊗ L^{2}[0, 1] is given, upon identifying this Hilbert space with L^{2}([0, 1]^{2}), by

h(s, t) = \begin{cases}1 & if s < t \\ 0 & if s ≥ t . \end{cases}

Suppose a ∈ N ∩ {T, T^{∗}}′ and τ(a) = 0. Since aX = Xa, it is well–known that we must have a ∈ W^{∗}(X), so â ∈ ℋ_{2}^{o}. Let ˆ1 = b_{0}, b_{1}, b_{2}, … be an orthonormal basis for L^{2}[0, 1]. Then

h = ∑_{i,j=0}^{∞} c_{ij}b_{i} ⊗ b_{j}

for some c_{ij} ∈ C. Clearly c_{ij} ≠ 0 for some i, j > 0. Thus

T̂ = c_{00}X̂ + ∑_{i>0} c_{i0}b_{i} ⊗ X̂ + ∑_{j>0} c_{0j}X̂ ⊗ b_{j} + ∑_{i,j>0} c_{ij}b_{i} ⊗ X̂ ⊗ b_{j}
(aT − T a) = v + ∑_{i,j>0} c_{ij}(â ⊗ b_{i} ⊗ X̂ ⊗ b_{j} − b_{i} ⊗ X̂ ⊗ b_{j} ⊗ â) ,

where

v ∈ C ⊕ ℋ_{2}^{o} ⊕ (ℋ_{1}^{o} ⊗ ℋ_{2}^{o}) ⊕ (ℋ_{2}^{o} ⊗ ℋ_{1}^{o}) ⊕ (ℋ_{2}^{o} ⊗ ℋ_{1}^{o} ⊗ ℋ_{2}^{o}) .

Since a commutes with T, we conclude that â = 0, i.e. a = 0. □

As remarked above, we will let Q denote the von Neumann subalgebra of N generated by T = 𝒰𝒯(X, λ). It is an interesting question whether the von Neumann algebra Q generated by T contains any nontrivial invariant projections for T, i.e. any projections p neither 0 nor 1 such that T p = pT p. Clearly, if Q is all of N, then many such projections exist. Moreover, if there is one such projection, then by compressing one can find a coninuum of them, that together densely span a diffuse abelian subalgebra of Q. Finally, using this idea we see that if λ(1_{[0,t]}) ∈ Q for some 0 < t < 1 then Q = N.

Lemma 6.2. Let μ be a compactly supported Borel measure on C. Then there is f ∈ L^{∞}[0, 1] whose distribution is μ and such that, if D = λ(f) and if T = 𝒰𝒯(X, λ) is as above, then for any c > 0, D itself lies in the W^{∗}–algebra generated by D + cT.

Proof. Simply take f ∈ L^{∞}([0, 1]) whose distribution is μ and that satisfies the following:

(i) for every atom a of μ, f^{−1}({a}) is a half–open interval [c, d); (ii) if a_{i} (i ∈ I) are the atoms of μ then the restriction of f to

[0, 1]\f^{−1}({a_{i} | i ∈ I})

is a measure preserving isomorphism of measure spaces. Let Z = D + cT and let W^{∗}(Z) denote the von Neumann subalgebra of N generated by Z. If t ∈ [0, 1] is such that f^{−1}(f([0, t))) = [0, t) then letting B = f([0, t)), and using Lemma 4.10 and Theorem 5.6, we get p_{Z}(B) = λ(1_{[0,t)}) ∈ W^{∗}(Z). Since p_{Z}(B) ∈ W^{∗}(Z), it follows that D ∈ W^{∗}(Z). □

Using Lemma 6.2 we see that if the von Neumann algebra Q is in fact all of N, then every DT–operator can be embedded in N in such a way that it generates all of N, which is isomorphic to L(F_{2}).

Proposition 6.3. If Z is a DT(μ, c)–operator, then the von Neumann algebra generated by Z is a II_{1}–factor whose isomorphism class depends only on μ, and in fact, only on the number and sizes of the atoms of μ.

Proof. Using Lemma 6.2 and Theorem 4.4, we see that the isomorphism class of the von Neumann algebra generated by a DT(μ, c)–operator depends only on the isomorphism class of the von Neumann algebra generated by a normal element whose distribution is μ. □

Theorem 6.4. If μ has no atoms and c > 0 then a DT(μ, c)–operator generates a von Neumann algebra isomorphic to L(F_{2}).

Proof. In the proof of Lemma 6.2, we see that D generates all of λ(L^{∞}([0, 1])) and the constructed DT-operator generates all of N ≅ L(F_{2}). □

7. DT–operators that are also R–diagonal

In [6] we introduced the circular free Poisson elements of parameters c ≥ 1, which are R–diagonal elements and (see Theorem 3.1) also DT–elements. In this section we will show that, up to multiplication with scalars, these are the only DT–elements that are also R–diagonal.

Let μ be a Borel probability measure compactly supported in C. For every n ∈ N, let D(n) ∈ ℳ_{n} be a diagonal random matrix whose entries are independent, each with distribution μ, and let T (n) ∈ UTGRM(n, \frac{1}{n}) be such that D(n) and T (n) are independent as matrix–valued random variable. From Theorem 2.1, (see also Remarks 2.3 and 4.8) it follows that the pair (D(n), T (n)) converges in ∗–moments as n → ∞ to a pair (D, T) in a tracial W^{∗}–noncommutative probability space (M, τ). Of course, by Definition 2.2, D + cT is a DT(μ, c)–element.

In the lemmas to follow, orthogonality and the 2–norm, ∥ · ∥_{2}, will be with respect to the trace τ.

Lemma 7.1. (i) For all p ∈ N ∪ {0},

∥(DT)^{p}D∥_{2}^{2} = \frac{∥D∥_{2}^{2(p+1)}}{(p + 1)!}

(ii) If p, q ∈ N ∪ {0} and p ≠ q, then (DT)^{p}D and (DT)^{q}D are orthogonal.

Proof. By the proof of Theorem 2.1, τ((DT)^{p}DD^{∗}(T^{∗}D^{∗})^{q}) is equal to the quantity (16), where

ϵ(1), …, ϵ(p + q) is \underbrace{1, …, 1}_{p}, \underbrace{∗, …, ∗}_{q}

and where E(σ; ⋯) must be properly interpreted.

If p ≠ q then there are no noncrossing pairings of {1, …, p + q} compatible with ϵ(1), …, ϵ(p + q), so τ((DT)^{p}DD^{∗}(T^{∗}D^{∗})^{q}) = 0 and (ii) is proved.

If p = q, then there exactly only one noncrossing pairing σ of {1, …, 2p} compatible with ϵ(1), …, ϵ(2p), namely σ = {{1, 2p}, {2, 2p − 1}, …, {p, p + 1}}. Following Algorithm 2.5, we find that the quotient graph Q is the straight line having 2p edges all oriented in the same direction and hence NTO(σ; ϵ(1), …, ϵ(2p)) = 1. Furthermore, the quantity E(σ; ⋯), which is given by equation (15), is equal to

( ∫_{C} |z|^{2}dμ(z))^{p+1} .
□

Lemma 7.2. Let M_{μ}(k, ℓ) = ∫_{C} z^{k}\overline{z}^{ℓ}dμ(z). Then for every |λ| < (∥D∥ + ∥T ∥)^{−1}, we have

∥ ∑_{n=0}^{∞} λ^{n}(D + T)^{n}∥_{2}^{2} = \frac{1}{|λ|^{2}} ( exp ( ∑_{k,ℓ=0}^{∞} λ^{k+1}(\overline{λ})^{ℓ+1}M_{μ}(k, ℓ)) − 1)

Proof. Because 1 − λD is invertible and

∥(1 − λD)^{−1}λT ∥ ≤ \frac{|λ|∥T ∥}{1 − |λ|∥D∥} < 1,

we have

∑_{n=0}^{∞} λ^{n}(D + T)^{n} = (1 − λ(D + T))^{−1} = ((1 − λD) − λT)^{−1}

= (1 − (1 − λD)^{−1}λT)^{−1}(1 − λD)^{−1}

= ∑_{k=0}^{∞}((1 − λD)^{−1}λT)^{k}(1 − λD)^{−1} .

Now applying Lemma 7.1 to (1 − λD)^{−1} instead of D gives

∥ ∑_{n=0}^{∞} λ^{n}(D + T)^{n}∥_{2}^{2} = ∑_{k=0}^{∞} \frac{|λ|^{2k}}{(k + 1)!}∥(1 − λD)^{−1}∥_{2}^{2(k+1)}

= \frac{exp(|λ|^{2}∥(1 − λD)^{−1}∥_{2}^{2}) − 1}{|λ|^{2}} .

This finishes the proof, since

|λ|^{2}∥(1 − λD)^{−1}∥_{2}^{2} = |λ|^{2}τ((1 − λD)^{−1}(1 − \overline{λ}D^{∗})^{−1})

= |λ|^{2} ∑_{k,ℓ=0}^{∞} λ^{k}(\overline{λ})^{ℓ}M_{μ}(k, ℓ) .

□

Theorem 7.3. Let Z be a DT(μ, 1)–element. Then Z is R–diagonal if and only if μ is the uniform distribution on the annulus

\{λ ∈ C | \sqrt{c − 1} ≤ |λ| ≤ \sqrt{c}\}  (34)

for some c ≥ 1, in which case Z is a circular free Poisson element of parameter c.

Proof. If μ is the uniform distribution on the annulus (34), then by Theorem 3.1 μ is circular free Poisson and is therefore R–diagonal.

Suppose Z is an R–diagonal element in a W^{∗}–noncommutative probability space (M, τ). Again, 2–norms ∥·∥_{2} will be with respect to τ. We may take Z = D+T with D, T ∈ M as defined at the beginning of this section. By [10], R–diagonality of Z implies ∥Z^{n}∥_{2}^{2} = ∥Z∥_{2}^{2n} for every n ∈ N. Since Z and ζZ have the same ∗–moments for all ζ ∈ T, we also have τ(Z^{n}(Z^{∗})^{m}) = 0 whenever n ≠ m. Hence for

|λ| < (∥D∥ + ∥T ∥)^{−1} ≤ ∥Z∥^{−1}

we have

∥ ∑_{n=0}^{∞} λ^{n}Z^{n}∥_{2}^{2} = \frac{1}{1 − |λ|^{2}∥Z∥_{2}^{2}} ,

so by Lemma 7.2,

\frac{1}{|λ|^{2}} ( exp ( ∑_{k,ℓ=0}^{∞} λ^{k+1}(\overline{λ})^{ℓ+1}M_{μ}(k, ℓ)) − 1) = \frac{1}{1 − |λ|^{2}∥Z∥_{2}^{2}} ,

from which we get

∑_{k,ℓ=0}^{∞} λ^{k+1}(\overline{λ})^{ℓ+1}M_{μ}(k, ℓ) = log (\frac{1 − |λ|^{2}(∥Z∥_{2}^{2} − 1)}{1 − |λ|^{2}∥Z∥_{2}^{2}}) .

Since the RHS has a power series expansion in |λ|^{2}, we must have M_{μ}(k, ℓ) = 0 whenver k ≠ ℓ. Thus μ is rotationally invariant. Let c = ∥Z∥_{2}^{2}. Substituting t = |λ|^{2} gives

∑_{k=0}^{∞} t^{k+1}M_{μ}(k, k) = log(1 − (c − 1)t) − log(1 − ct)

= ∑_{j=1}^{∞} (\frac{(ct)^{j}}{j} − \frac{((c − 1)t)^{j}}{j} )

= ∑_{k=0}^{∞} \frac{c^{k+1} − (c − 1)^{k+1}}{k + 1} t^{k+1} ,

which implies that

τ((DD^{∗})^{k}) = M_{μ}(k, k) = \frac{c^{k+1} − (c − 1)^{k+1}}{k + 1} t^{k+1} = ∫_{c−1}^{c} x^{k}dx .

Thus the distribution of DD^{∗} is the uniform distribution on the interval [c − 1, c]. In particular, c ≥ 1 since DD^{∗} is a positive operator, and μ is uniform measure on the annulus (34). □

8. The distribution of T^{∗}T

Throughout this section, (A, τ) will be a ∗–noncommutative probability space and T will denote a DT(δ_{0}, 1)–element in (A, τ). We will investigate the moments of the element T^{∗}T, finding its distribution and R–transform.

Let 1 ∈ B ⊆ A be a unital ∗–subalgebra. The following result is standard.

Proposition 8.1. If u ∈ A is a unitary such that τ(u) = 0 and B and {u} are ∗–free, then B and u^{∗}Bu are free.

Corollary 8.2. Suppose z ∈ A is a circular element such that B and {z} are ∗–free. Let D be the ∗–subalgebra of A generated by {z^{∗}bz | b ∈ B}. Then B and D are free.

Proof. By Voiculescu’s result [24, Proposition 2.6] about the polar decomposition of a circular element, enlarging A if necessary we may without loss of generality assume z = hu where u is a Haar unitary, h ≥ 0 and u and h are ∗–free. Then D ⊆ u^{∗}Cu, where C is the ∗–subalgebra of A generated by B ∪ {h}. Since C and {u} are ∗–free, it follows from Proposition 8.1 that D and C are free, and therefore that D and B are free. □

Lemma 8.3. Suppose N ∈ N and (a_{ij})_{1≤i≤j≤N} is a ∗–free family of circular elements in (A, τ) such that τ(a_{ij}^{∗}a_{ij}) is the same for all i and j. In the ∗–noncommutative probability space (M_{N}(A), τ ∘ tr_{N}), consider the elements

R = (\begin{matrix}0 & 0 & 0 & ⋯ & 0 \\ 0 & a_{22} & a_{23} & ⋯ & a_{2N} \\ 0 & 0 & a_{33} & ⋯ & a_{3N} \\ ⋮ & ⋮ & ... & ... & ⋮ \\ 0 & 0 & ⋯ & 0 & a_{NN} \end{matrix} ) , Q = (\begin{matrix}a_{11} & a_{12} & ⋯ & a_{1N} \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) .

Then R and Q^{∗}Q are ∗–free with respect to τ ∘ tr_{N}.

Proof. Expanding A if necessary, let b_{kℓ}, c_{rs} ∈ A be such that

(a_{ij})_{1≤i≤j≤N,} (b_{kℓ})_{1≤k,ℓ≤N,} (c_{rs})_{1≤r,s≤N}

is a ∗–free family of circular elements in (A, τ) such that τ(a_{ij}^{∗}a_{ij}) = τ(b_{kℓ}^{∗}b_{kℓ}) = τ(c_{rs}^{∗}c_{rs}) for all i, j, k, ℓ, r and s. Let

Y = (\begin{matrix}b_{11} & b_{12} & b_{13} & ⋯ & b_{1N} \\ b_{21} & a_{22} & a_{23} & ⋯ & a_{2N} \\ b_{31} & b_{32} & a_{33} & ⋯ & a_{3N} \\ ⋮ & ⋮ & ... & ... & ⋮ \\ b_{N1} & b_{N2} & ⋯ & b_{N−1,N} & a_{NN} \end{matrix} ) , Z = (\begin{matrix}a_{11} & a_{12} & ⋯ & a_{1N} \\ c_{21} & c_{22} & ⋯ & c_{2N} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ c_{N1} & c_{N2} & ⋯ & c_{NN} \end{matrix} ) .

Moreover, for 1 ≤ j ≤ N let p_{j} ∈ M_{N}(A) be the diagonal N × N matrix with 1 in the jth diagonal entry and zeros elsewhere. Then by results of Voiculescu [24], with respect to τ ∘ tr_{N}, Y and Z are circular and the family

{Y }, {Z}, {p_{j} | 1 ≤ j ≤ N}

of sets of random variables is ∗–free. Since R belongs to the subalgebra of A generated by {Y } ∪ {p_{j} | 1 ≤ j ≤ N} and since Q = p_{1}Z, ∗–freeness of R and Q^{∗}Q follows from Corollary 8.2. □

Lemma 8.4. In a W^{∗}–noncommutative probability space (A, τ) let (Y_{ij})_{1≤i≤j≤N} be a ∗–free family of circular elements with τ(Y_{ij}^{∗}Y_{ij}) = 1. For N ∈ N, consider the elements

S_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}0 & Y_{12} & Y_{13} & ⋯ & Y_{1N} \\ 0 & 0 & Y_{23} & ⋯ & Y_{2N} \\ 0 & ... & ... & ... & ⋮ \\ ⋮ & ... & 0 & 0 & Y_{N−1,N} \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix})

T_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}Y_{11} & Y_{12} & Y_{13} & ⋯ & Y_{1N} \\ 0 & Y_{22} & Y_{23} & ⋯ & Y_{2N} \\ 0 & ... & ... & ... & ⋮ \\ ⋮ & ... & 0 & ... & Y_{N−1,N} \\ 0 & ⋯ & 0 & 0 & Y_{N,N} \end{matrix} )

in the W^{∗}–noncommutative probability space (A ⊗ M_{N}(C), τ_{N}), where τ_{N} = τ ⊗ tr_{N} with tr_{N} the normailzed trace on M_{N}(C). Then

(a) for all p, N ∈ N with N ≥ 2,

τ_{N}((S_{N}^{∗}S_{N})^{p} = (\frac{N − 1}{N})^{p+1}τ_{N−1}((T_{N−1}^{∗}T_{N−1})^{p}),

(b) if ν_{N} and μ_{N} denote the distributions of T_{N}^{∗}T_{N} and S_{N}^{∗}S_{N}, respectively, computed with respect to τ_{N}, then their R–transforms satisfy

R_{ν_{N}} (z) = R_{μ_{N}} (z) + \frac{1}{N(1 − z)}

for all z in a neighborhood of 0 in C.

Proof. It is clear that S_{N} has the same ∗–moments as

S′_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}0 & Y_{11} & Y_{12} & ⋯ & Y_{1,N−1} \\ 0 & 0 & Y_{22} & ⋯ & Y_{2,N−1} \\ 0 & ... & ... & ... & ⋮ \\ ⋮ & ... & 0 & 0 & Y_{N−1,N−1} \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) .

Moreover,

(S′_{N})^{∗}S′_{N} = \frac{N − 1}{N} (\begin{matrix}0 & 0 ⋯ 0 \\ 0 \\ ⋮ & T_{N−1}^{∗}T_{N−1} \\ 0 \end{matrix}) .

Thus

τ_{N}((S_{N}^{∗}S_{N})^{p}) = τ_{N}(((S′_{N})^{∗}S′_{N})^{p}) = (\frac{N − 1}{N})^{p+1}τ_{N−1}((T_{N−1}^{∗}T_{N−1})^{p}),

which proves (a).

For (b), write T_{N} = P_{N} + Q_{N}, where

P_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}0 & 0 & ⋯ & 0 \\ 0 & Y_{22} & ⋯ & Y_{2N} \\ 0 & ... & ⋮ \\ 0 & ⋯ & 0 & Y_{N,N} \end{matrix} ) , Q_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}Y_{11} & Y_{12} & ⋯ & Y_{1N} \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) .

Since P_{N}^{∗}Q_{N} = 0, we have

T_{N}^{∗}T_{N} = P_{N}^{∗}P_{N} + Q_{N}^{∗}Q_{N}.

By Lemma 8.3, P_{N}^{∗}P_{N} and Q_{N}^{∗}Q_{N} are free with respect to τ_{N}. Moreover, P_{N}^{∗}P_{N} = (S′′_{N})^{∗}S′′_{N}, where

S′′_{N} = \frac{1}{\sqrt{N}} (\begin{matrix}0 & Y_{22} & Y_{23} & ⋯ & Y_{2N} \\ 0 & 0 & Y_{33} & ⋯ & Y_{3N} \\ 0 & ... & ... & ... & ⋮ \\ ⋮ & ... & 0 & 0 & Y_{NN} \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) .

Note that S′′_{N} has the same ∗–moments as S_{N}. Hence P_{N}^{∗}P_{N} has the same distribution as S_{N}^{∗}S_{N}. By [26, Theorem 1.6], the distribution of Q_{N}^{∗}Q_{N} with respect to τ_{N} is free Poisson of parameter 1/N. In particular, (cf. [27, p. 34] or [13, Example 3.3.5]), if σ is the distribution of Q_{N}^{∗}Q_{N} computed with respect to τ_{N}, then

R_{σ}(z) = \frac{1}{N(1 − z)}.

This proves part (b). □

Lemma 8.5. Let N ∈ N, let α_{N}(0) = β_{N}(0) = 1 and for p ∈ N let

α_{N}(p) = \frac{(p − \frac{1}{N})(p − \frac{2}{N}) ⋯ (p − \frac{p}{N})}{(p + 1)!}
β_{N}(p) = \frac{(p + \frac{1}{N})(p + \frac{2}{N}) ⋯ (p + \frac{p}{N})}{(p + 1)!} .

Then (a) the function

K_{N}(t) = \frac{t}{1 − ∑_{p=0}^{∞} α_{N}(p)t^{p+1}}

is defined, analytic and invertible with respect to composition in a neighborhood of 0 in C, and its inverse is given by

K_{N}^{⟨−1⟩}(z) = z(1 + \frac{z}{N})^{−N}

in some neighborhood of 0 in C;

(b) the function

L_{N}(t) = \frac{t}{1 − ∑_{p=0}^{∞} β_{N}(p)t^{p+1}}

is defined, analytic and invertible with respect to composition in a neighborhood of 0 in C, and its inverse is given by

L_{N}^{⟨−1⟩}(z) = z(1 − \frac{z}{N})^{N}

in some neighborhood of 0 in C.

Proof. For (a), let g(z) = z(1 + \frac{z}{N})^{−N}, z ∈ C\\{−N\}. Since g(0) = 0 and g′(0) = 1, g is invertible with respect to composition in a neighborhood of 0. We must show that

g^{⟨−1⟩}(t) = \frac{t}{1 − ∑_{p=0}^{∞} α_{N}(p)t^{p+1}}

in some neighborhood of 0 in C, which is equivalent to showing that

\frac{t}{g^{⟨−1⟩}(t)} = 1 − ∑_{p=0}^{∞} α_{N}(p)t^{p+1}  (35)

in some neighborhood of 0. Since t/g^{⟨−1⟩}(t) is analytic in a neighborhood of 0, it is of the form

\frac{t}{g^{⟨−1⟩}(t)} = ∑_{p=0}^{∞} a_{p}t^{p}, 0 < |t| < r

for some r > 0. Since g(z) = z − z^{2} + O(|z|^{3}) as z → 0, one has

g^{⟨−1⟩}(t) = t + t^{2} + O(|t|^{3})
\frac{t}{g^{⟨−1⟩}(t)} = 1 − t + O(|t|^{2})

as t → 0. Therefore, a_{0} = 1 and a_{1} = −1 = −α_{N}(0).

For p ≥ 2, we use the formula

a_{p} = \frac{1}{2πi} ∫_{C} \frac{t}{g^{⟨−1⟩}(t)t^{p+1}} dt,  (36)

where C is the positively oriented path around a circle centered at 0 with radius less than r. If the radius of C is chosen small enough, then the image, C′, of C under g^{⟨−1⟩} will be a smooth, simple, closed curve winding once around 0 in the positive direction. The substitution t = g(z) in the integral (36) yields

a_{p} = \frac{1}{2πi} ∫_{C′} \frac{g′(z)}{zg(z)^{p}} dz.

Since

d(g(z)^{1−p}) = (1 − p)\frac{g′(z)}{g(z)^{p}} dz,

integrating by parts yields

(1 − p)a_{p} = \frac{1}{2πi} ∫_{C′} \frac{1}{z}d(g(z)^{1−p})  = −\frac{1}{2πi} ∫_{C′} g(z)^{1−p}d(\frac{1}{z})
= \frac{1}{2πi} ∫_{C′} \frac{g(z)^{1−p}}{z^{2}} dz  = \frac{1}{2πi} ∫_{C′} \frac{(1 + \frac{z}{N})^{N(p−1)}}{z^{p+1}} dz
= Res(\frac{(1 + \frac{z}{N})^{N(p−1)}}{z^{p+1}} ; z = 0) = \binom{N(p − 1)}{p}\frac{1}{N^{p}} ,

where the last quantity is the coefficient of z^{p} in the polynomial expansion of (1 + \frac{z}{N})^{N(p−1)}. Thus for p ≥ 1,

a_{p+1} = −\frac{1}{p}\binom{Np}{p + 1}\frac{1}{N^{p+1}} = −\frac{(Np − 1)(Np − 2) … (Np − p)}{(p + 1)! N^{p}} = −α_{N}(p).

We have shown a_{0} = 1 and a_{p+1} = −α_{N}(p) for all p ≥ 0, which proves equation (35) and finishes the proof of part (a).

Part (b) follows by minor modifications of the above proof. Put

h(z) = z(1 − \frac{z}{N})^{N}

and let

\frac{t}{h^{⟨−1⟩}(t)} = ∑_{p=0}^{∞} b_{p}t^{p}

be the power series expansion, which is valid for 0 < |t| < s for some s > 0. As before, one finds b_{0} = 1, b_{1} = −1 = −β_{N}(0) and, for p ≥ 2,

(1 − p)b_{p} = Res(\frac{(1 − \frac{z}{N})^{−N(p−1)}}{z^{p+1}} ; z = 0) = \binom{−N(p − 1)}{p}\frac{(−1)^{p}}{N^{p}} ,

where the latter quantity is the coefficient of z^{p} in the power series expansion of (1 − \frac{z}{N})^{−N(p−1)}. Thus, for p ≥ 1,

b_{p+1} = \frac{(−1)^{p}}{pN^{p+1}} \binom{−Np}{p + 1} = −\frac{(Np + 1)(Np + 2) … (Np + p)}{(p + 1)! N^{p}} = −β_{N}(p),

which shows that

\frac{t}{h^{⟨−1⟩}(t)} = 1 − ∑_{p=0}^{∞} β_{N}(p)t^{p+1}

for 0 < |t| < s. This proves part (b). □

Lemma 8.6. Let S_{N}, T_{N}, τ_{N}, μ_{N} and ν_{N} be as in Lemma 8.4, (N ∈ N). Then (i) for all p ∈ N,

τ_{N}((S_{N}^{∗}S_{N})^{p}) = \frac{(p − \frac{1}{N})(p − \frac{2}{N}) ⋯ (p − \frac{p}{N})}{(p + 1)!} ,

(ii) for all p ∈ N,

τ_{N}((T_{N}^{∗}T_{N})^{p}) = \frac{(p + \frac{1}{N})(p + \frac{2}{N}) ⋯ (p + \frac{p}{N})}{(p + 1)!},

(iii) for all z in a neighborhood of 0 in C,

R_{μ_{N}} (z) = \frac{1}{N(1 − z)((1 − z)^{−1/N} − 1)} − \frac{1}{z},

(iv) for all z in a neighborhood of 0 in C,

R_{ν_{N}} (z) = \frac{1}{N(1 − z)(1 − (1 − z)^{1/N})} − \frac{1}{z}.

Proof. We proceed by induction on N. We have S_{1}^{∗}S_{1} = 0, so (i) and (iii) hold when N = 1. Moreover, (T_{1}^{∗}T_{1})^{1/2} = (Y_{11}^{∗}Y_{11})^{1/2} is a quarter–circular element, hence

τ_{N}((T_{1}^{∗}T_{1})^{p}) = \frac{1}{π} ∫_{0}^{2} x^{2p}\sqrt{4 − x^{2}}dx = \frac{1}{p + 1}\binom{2p}{p},

(the pth Catalan number), which proves (ii) when N = 1. But T_{1}^{∗}T_{1} has the free Poisson distribution with parameter 1, which implies that R_{ν_{N}} = 1/(1 − z), proving (iv) when N = 1.

Now suppose N ≥ 2. By the induction hypothesis, for every p ∈ N we have

τ_{N−1}((T_{N−1}^{∗}T_{N−1})^{p}) = \frac{(p + \frac{1}{N−1})(p + \frac{2}{N−1}) … (p + \frac{p}{N−1})}{(p + 1)!}.

Then by part (a) of Lemma 8.4,

τ_{N}((S_{N}^{∗}S_{N})^{p}) = (\frac{N − 1}{N})^{p+1} \frac{(p + \frac{1}{N−1})(p + \frac{2}{N−1}) … (p + \frac{p}{N−1})}{(p + 1)!}

= (\frac{N − 1}{N^{p+1}})\frac{((N − 1)p + 1)((N − 1)p + 2) … ((N − 1)p + (p − 1))Np}{(p + 1)!}

= \frac{(pN − p)(pN − (p − 1)) … (pN − 1)}{N^{p}(p + 1)!}

= \frac{(p − \frac{p}{N})(p − \frac{p−1}{N}) … (p − \frac{1}{N})}{(p + 1)!}.

This proves (i) for this particular N.

We now show (i) ⟹ (iii). By (i), the Cauchy transform of μ_{N} is given by

G_{μ_{N}} (λ) = ∑_{p=0}^{∞} α_{N}(p)λ^{−p−1}

for λ ∈ C with |λ| large, where α_{N}(p) is as defined in Lemma 8.5. Thus

K_{N}(t) = \frac{t}{1 − G_{μ_{N}} (1/t)}

for t in some neighborhood of 0, where K_{N} is as defined in Lemma 8.5. Since by Lemma 8.5,

K_{N}^{⟨−1⟩}(w) = w(1 + \frac{w}{N})^{−N}

in a neighborhood of 0, we have for |w| small

w = K_{N}(K_{N}^{⟨−1⟩}(w)) = \frac{w(1 + \frac{w}{N})^{−N}}{1 − G_{μ_{N}} (\frac{1}{w}(1 + \frac{w}{N})^{N})}.

Hence for |w| small,

G_{ν_{N}} (\frac{1}{w}(1 + \frac{w}{N} )^{N}) = 1 − (1 + \frac{w}{N} )^{−N}.  (37)

Letting z = 1 − (1 + \frac{w}{N})^{−N} for small values of |w|, we have w = N((1 − z)^{−1/N} − 1) and (1 + \frac{w}{N})^{N} = 1/(1 − z). Thus for small values of |z|, we have

G_{ν_{N}} (\frac{1}{N((1 − z)^{−1/N} − 1)(1 − z)}) = z.

Therefore

R_{μ_{N}} (z) = G_{μ_{N}}^{⟨−1⟩}(z) − \frac{1}{z} = \frac{1}{N(1 − z)((1 − z)^{−1/N} − 1)} − \frac{1}{z},

which proves (iii) for this particular N.

Part (iii), together with part (b) of Lemma 8.4, proves (iv) for this particular N. Finally, we prove (ii) (still for this N). Since there is a one–to–one correspondence

between moment series and R–series, it will suffice to prove that if (ii) holds, then the corresoponding R–transform R_{ν_{N}} is given by (iv). This in turn follows by minormodifications of the above proof of (i) ⟹ (iii). Indeed, if (ii) holds then the Cauchy transform of ν_{N} is

G_{ν_{N}} (λ) = ∑_{p=0}^{∞} β_{N}(p)λ^{−p−1}.

Thus

L_{N}(t) = \frac{t}{1 − G_{ν_{N}} (1/t)}

for t in a neighborhood of 0, where L_{N} is the function defined in Lemma 8.5. Thus

L_{N}^{⟨−1⟩}(w) = w(1 − \frac{w}{N} )^{N}

for |w| small, and arguing as in the proof of (i) ⟹ (iii) shows

G_{ν_{N}} (\frac{1}{w}(1 − \frac{w}{N} )^{−N}) = 1 − (1 − \frac{w}{N} )^{N}

for |w| small. Setting z = 1 − (1 − \frac{w}{N})^{N} yields w = N(1 − (1 − z)^{1/N}) and

G_{ν_{N}} (\frac{1}{N(1 − (1 − z)^{1/N})(1 − z)}) = z,

proving (iv), as desired. This concludes the proof of the induction step, and of the lemma. □

Recall that T is a DT(δ_{0}, 1)–operator in a W^{∗}–noncommutative probability space (A, τ).

Theorem 8.7. (a) For every p ∈ N,

τ((T^{∗}T)^{p}) = \frac{p^{p}}{(p + 1)!}.

(b) If ν is the distribution of T^{∗}T, then its R–transform is given by

R_{ν}(z) = \frac{−1}{(1 − z) log(1 − z)} − \frac{1}{z}

for |z| small.

Proof. Without loss of generality assume (A, τ) is a W^{∗}–noncommutative probability space with τ faithful. By Theorem 4.12, enlarging (A, τ) if necessary, S_{N} differs from a DT (δ_{0}, 1)–operator by a diagonal matrix

A = diag(A_{1}, …, A_{N}) ∈ A ⊗ M_{N}(C),

where each A_{j} is DT(δ_{0}, 1/\sqrt{N}). Thus ∥A∥ ≤ 2/\sqrt{N}. Also, T_{N} differs from S_{N} by a diagonal matrix of norm 2/\sqrt{N}, and we conclude that T_{N} converges in ∗–moments to T as N → ∞. Taking the limit as N → ∞ of the value of τ_{N}((T_{N}^{∗}T_{N})^{p}) given in part (ii) of Lemma 8.6 yields (a) above.

The convergence in ∗–moments of T_{N} to T ensures that the coefficients of the R– series of the distribution ν_{N} of T_{N}^{∗}T_{N} converge to the coefficients of the R–series of the distribution ν of T^{∗}T. Writing (1 − z)^{1/N} = exp(log(1 − z)/N) and taking the limit as N → ∞ of the formula in part (iv) of Lemma 8.6 proves part (b). □

Lemma 8.8. Let γ(0) = 1 and for p ∈ N let

γ(p) = \frac{p^{p}}{(p + 1)!}.

Then the function

L(t) = \frac{t}{1 − ∑_{p=0}^{∞} γ(p)t^{p+1}}

is defined, analytic and invertible with respect to composition in a neighborhood of 0 in C, and its inverse is given by

L^{⟨−1⟩}(z) = ze^{−z}  (38)

for |z| small.

Proof. It is clear that L is defined and analytic in a neighborhood of 0. Since L(0) = 0 and L′(0) = 1, L is invertible in a neighborhood of 0. Turning now to Lemma 8.5, since lim_{N→∞} β_{N}(p) = γ(p) for all p ∈ N ∪ {0}, it follows that the coefficients in the power series expansion for L_{N}(t) around 0 converge as N → ∞ termwise to the ceoffiecients in the power series expansion of L(t). Hence the coefficients in the power series for L_{N}^{⟨−1⟩}(z) converge to those in the power series expansion L^{⟨−1⟩}(z). Since L_{N}^{⟨−1⟩}(z) = z(1 − \frac{z}{N})^{N}, we get the formula (38). □

Theorem 8.9. The distribution ν of T^{∗}T is absolutely continuous with respect to Lebesgue measure and has support equal to the interval [0, e]. Its density function ϕ is defined on the interval (0, e) by

ϕ(\frac{sin v}{v} exp(v cot v)) = \frac{1}{π} sin v exp(−v cot v),  0 < v < π.  (39)

Proof. By part (a) of Theorem 8.7, the Cauchy transform of ν is

G_{ν}(λ) = ∑_{p=0}^{∞} γ(p)λ^{−p−1}.

Using Lemma 8.8 and the technique used in the proof of equation (37) in Lemma 8.6, we find

G_{ν}(w^{−1}e^{w}) = 1 − e^{−w}

for |w| small. Define ρ : (0, π) → R by

ρ(v) = \frac{sin v}{v} exp(v cot v).  (40)

Then ρ(v) > 0 and ρ is strictly decreasing, because for 0 < v < π,

\frac{d}{dv} log ρ(v) = 2 cot v − \frac{1}{v} − \frac{v}{sin^{2} v}

= −(\frac{(v − sin v)^{2} + 2v sin v(1 − cos v)}{v sin^{2} v}) < 0.

Moreover,

lim_{v→0^{+}} ρ(v) = e,  lim_{v→π^{−}} ρ(v) = 0,

so ρ is an order–reversing bijection from (0, π) onto (0, e). Hence we may define

ϕ : (0, e) → (0, ∞)

by equation (39). Using substitution and integration by parts, we have

∫_{0}^{e} ϕ(x)dx = − ∫_{0}^{π} ϕ(ρ(v))ρ′(v)dv

= − ∫_{0}^{π} \frac{1}{π} sin v exp(−v cot v)d(\frac{sin v}{v} exp(v cot v))

= \frac{− sin^{2} v}{πv} |_{0}^{π} + \frac{1}{π} ∫_{0}^{π} \frac{sin v}{v} exp(v cot v)d( sin v exp(−v cot v))

= \frac{1}{π} ∫_{0}^{π} (\frac{sin v}{v} )(\frac{v}{sin v} )dv = 1.

So ϕ is the density of a probability measure on (0, e).

It remains to prove that for all p ∈ N,

∫_{0}^{e} x^{p}ϕ(x)dx = γ(p) = \frac{p^{p}}{(p + 1)!}.

Let g(w) = we^{−w}. By Lemma 8.8, the inverse of g in a neighborhood of 0 is

g^{⟨−1⟩}(t) = \frac{t}{1 − ∑_{p=0}^{∞} γ(p)t^{p+1}} .

As in the proof of Lemma 8.5, γ(p) can be recovered for p ≥ 1 as γ(p) = −c_{p+1} where

c_{p+1} = \frac{1}{2πi} ∫_{C′} \frac{g′(w)}{wg(w)^{p+1}} dw,

for some smooth simple closed path C′ winding once around 0 with positive orientation. In our case, g(w) = we^{−w} and g′(w) = (1 − w)e^{−w}, so

γ(p) = −c_{p+1} = \frac{1}{2πi} ∫_{C′} \frac{w − 1}{w^{p+2}}e^{pw}dw.  (41)

Since the integrand is analytic in C\{0}, we may choose C′ to be any piecewise smooth closed path winding once around 0 with positive orientation. Let α ∈ (\frac{π}{2}, π) and put C′ = C_{1}^{(α)} ∪ C_{2}^{(α)}, where C_{1}^{(α)} and C_{2}^{(α)} are parameterized by

C_{1}^{(α)} : v ↦ v cot v + iv,  −α ≤v ≤ α
C_{2}^{(α)} : v ↦ α cot α − iv, −α ≤v ≤ α,

setting v cot v = 1 when v = 0. Since α cot α → −∞ as α → π^{−}, it is clear that

| ∫_{C_{2}^{(α)}} \frac{w − 1}{w^{p+2}}e^{pw}dw| → 0

as α → π^{−}, for all p ≥ 1. Hence

γ(p) = \frac{1}{2πi} ∫_{C_{1}} \frac{w − 1}{w^{p+2}}e^{pw}dw, (p ∈ N),

where C_{1} is the path parametrized by

v ↦ w = v cot v + iv = \frac{v}{sin v} e^{iv}, −π < v < π,

with the convention v cot v = 1 = v/ sin v when v = 0. We have

(\frac{1}{w}e^{w})^{p} = (\frac{sin v}{v} exp(v cot v))^{p}

\frac{w − 1}{w^{2}}e^{w}dw = d(\frac{1}{w}e^{w}) = d(\frac{sin v}{v} exp(v cot v))
e^{−w} = e^{−v cot v−iv},

which give

γ(p) = \frac{1}{2πi} ∫_{−π}^{π} e^{−v cot v−iv}ρ(v)^{p}ρ′(v)dv,

where ρ is as defined in equation (40). Since ρ is an even function, ρ′ is odd and using equation (39) we get

γ(p) = \frac{−1}{2π} ∫_{−π}^{π} e^{−v cot v}(sin v)ρ(v)^{p}ρ′(v)dv

= \frac{−1}{π} ∫_{0}^{π}(sin v)e^{−v cot v}ρ(v)^{p}ρ′(v)dv = ∫_{0}^{e} ϕ(x)x^{p}dx,

as required. □

Remark 8.10. From Theorem 8.9 one gets

ϕ(x) ∼ \frac{1}{x(log x)^{2}}  as x → 0^{+}
ϕ(x) ∼ \frac{\sqrt{2}}{πe^{3/2}} (e − x)^{1/2} as x → e^{−}.

Note that the asymptotic behevior as x → 0^{+} implies

∫_{0}^{e} ϕ(x) log xdx = −∞.

This equality also follows from the fact that the Fuglede–Kadison determinant of any quasi–nilpotent operator in a II_{1}–factor is equal to 0.

Corollary 8.11. ∥T ∥ = \sqrt{e}.

Corollary 8.12. ker(T) = ker(T^{∗}T) = {0}.

Proof. This follows from ν({0}) = 0, where ν is the distribution of T^{∗}T. □

9. ∗–moments of the operator T

In this section, T will denote a DT(δ_{0}, 1)–element. Algorithm 2.5 allows computation of an arbitrary ∗–moment of T, though for large moments this becomes somewhat arduous. We will prove a recursion formula for general ∗–moments of T. The proof will use the construction T = 𝒰𝒯(X, λ), (see Definition 4.2), and will involve asymptotic behavior of certain products of free semicircular variables.

We begin with a few preparatory lemmas.

Lemma 9.1. In a W^{∗}–noncommutative probability space (A, τ), let B ⊆ A be a unital, commutative W^{∗}–subalgebra and let X̃ and Ỹ be semicircular elements having the same second moments and such that the family B, {X̃}, {Ỹ} is free. Let p ∈ B be a projection and let

X = pX̃p + pỸ(1 − p) + (1 − p)Ỹp + (1 − p)Ỹ(1 − p).

Then X is semicircular, has the same second moment as X̃ and Ỹ, and the pair {X}, B is free.

Proof. We may without loss of generality assume that τ is a faithful trace, B ≅ L^{∞}[0, 1] and τ(X̃^{2}) = 1 = τ(Ỹ^{2}). Let k ∈ N and let B_{k} ⊆ B be the linear span of a set of k orthogonal projections in B, each having trace 1/k. We will show that if p̃ ∈ B_{k} is a projection, 0 < τ(p̃) < 1, and if

Z = p̃X̃p̃ + p̃Ỹ(1 − p̃) + (1 − p̃)Ỹp̃ + (1 − p̃)Ỹ(1 − p̃),

then Z is semicircular with τ(Z^{2}) = 1 and the pair {Z}, B is free. By approximation, this will suffice to prove the lemma.

Let (D, τ_{D}) be a W^{∗}–noncommutative probability space having a ∗–free family (x_{ij})_{1≤j≤k}, (y_{ij})_{1≤j≤k} where for all i, x_{ii} and y_{ii} are semicircular elements with τ_{D}(x_{ii}^{2}) = τ_{D}(y_{ii}^{2}) = 1/k and for all i ≠ j, x_{ij} and y_{ij} are circular elements with τ_{D}(x_{ij}^{∗}x_{ij}) = τ_{D}(y_{ij}^{∗}y_{ij}) = 1/k. By results [25] of Voiculescu arising from his matrix model, we may take, in the W^{∗}–noncommutative probability space (M_{k}(D), τ_{D} ∘ tr_{k}), X̃ = (x_{ij})_{1≤i,j≤n}, Ỹ = (y_{ij})_{1≤i,j≤n}, B_{k} the set of diagonal k × k matrices with scalar entries and p̃ = diag(1, …, 1, 0, …, 0) with tr_{k}(p̃) = ℓ/k. Then

Z = (\begin{matrix}x_{11} & ⋯ & x_{1ℓ} & y_{1,ℓ+1} & ⋯ & y_{1k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ x_{ℓ1} & ⋯ & x_{ℓℓ} & y_{ℓ,ℓ+1} & ⋯ & y_{ℓk} \\ y_{ℓ+1,1} & ⋯ & y_{ℓ+1,ℓ} & y_{ℓ+1,ℓ+1} & ⋯ & y_{ℓ+1,k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ y_{k1} & ⋯ & y_{kℓ} & y_{k,ℓ+1} & ⋯ & y_{kk} \end{matrix}) .

By these same results in [25], Z is semicircular and the pair B_{k}, {Z} is free. □

Lemma 9.2. Let (A, τ) be a W^{∗}–noncommutative probability space, let B ⊆ A be a unital, commutative W^{∗}–subalgebra and let X ∈ A be a semicircular element such that the pair B, {X} is free. Let p ∈ B be a projection, 0 < τ(p) < 1. Then the family

pB, {pXp}, pW^{∗}({X(1 − p), (1 − p)X} ∪ (1 − p)B)p  (42)

is free with respect to τ(p)^{−1}τ↾_{pAp}.

Proof. Without loss of generality we may take X = pX̃p + pỸ(1 − p) + (1 − p)Ỹp + (1 − p)Ỹ(1 − p) with X̃, Ỹ and B as in Lemma 9.1 and with τ a faithful trace. Let D = Cp+C(1−p) and let E_{D} : A → D be the τ–preserving conditional expectation. Then the family

B, W^{∗}({X̃} ∪ D), W^{∗}({Ỹ} ∪ D)  (43)

is free over D with respect to E_{D}. Let us show that the family

pB, pW^{∗}({X̃} ∪ D)p, pW^{∗}({Ỹ} ∪ (1 − p)B)p  (44)

is free with respect to τ(p)^{−1}τ↾_{pAp}. Let

𝒴^{o} = W^{∗}({Ỹ} ∪ D) ∩ ker E_{D},
𝒳^{o} = W^{∗}({X̃} ∪ D) ∩ ker E_{D},
B^{o} = B ∩ ker E_{D}.

Let C = Cp + (1 − p)B and C^{o} = C ∩ B^{o}. By freeness of (43) over D, W^{∗}({Ỹ} ∪ (1 − p)B) ∩ ker E_{D} is the weak closure of the linear span of Λ^{o}(𝒴^{o}, C^{o}). (The definition of this notation can be found near equation (30).) From this and the fact that p is a minimal projection in C and D, we see that pW^{∗}({Ỹ} ∪ (1 − p)B)p ∩ ker τ is the weak closure of the linear span of pΘp, where Θ is the set of all words belonging to Λ^{o}(𝒴^{o}, C^{o}) whose first and last letters come from 𝒴^{o}. Therefore, to prove freeness of (44), it will suffice to show

Λ^{o}(pΘp, pW^{∗}({X̃} ∪ D)p ∩ ker τ, pB ∩ ker τ) ⊆ ker τ.

Since p is a minimal projection in D,

pW^{∗}({X̃} ∪ D)p ∩ ker τ ⊆ 𝒳^{o}.

Using this and the freeness of (43) over D, we have

Λ^{o}(pΘp, pW^{∗}({X̃} ∪ D)p ∩ ker τ, pB ∩ ker τ) ⊆
⊆ pΛ^{o}(Θ, pW^{∗}({X̃} ∪ D)p ∩ ker τ, pB ∩ ker τ)p
⊆ pΛ^{o}(𝒴^{o}, 𝒳^{o}, B^{o})p ⊆ p(ker E_{D})p ⊆ ker τ,

so the family (44) is free. However,

pXp ∈ pW^{∗}({X̃} ∪ D)p
(1 − p)X, (1 − p)X ∈ W^{∗}({Ỹ} ∪ (1 − p)B)

and the freeness of the family (42) follows from that of the family (44). □

Lemma 9.3. In a noncommutative probability space (Q, τ), let X be a semicircular element satisfying τ(X^{2}) = 1 and for n ∈ N define the function h_{n} : (0, 1) → [0, ∞] by

h_{n}(t) = τ((Xp_{t}X(1 − p_{t}))^{n}),

where p_{t} ∈ A is an idempotent such that τ(p_{t}) = t and the pair X, p_{t} is free. Then the asymptotic behavior of h_{n}(t) as t → 0 is

h_{n}(t) = t + O(t^{2}).  (45)

Proof. We will use Speicher’s moment–cummulant formula [22, Theorem 2.17], (see alternatively equation (3.14) of [18]), which gives

τ((Xp_{t}X(1 − p_{t}))^{n}) = ∑_{π∈NC(2n)} k_{π}[X, X, …, X]ϕ_{K(π)}[p_{t}, 1 − p_{t}, …, p_{t}, 1 − p_{t}]. (46)

Here the sum is over all non–crossing partitions π of {1, …, 2n}, k_{π}[X, …, X] is the corresponding cummulant and K(π) ∈ NC(2n) is the Kreweras complement of π. Since

k_{m}[X, …, X] = \begin{cases}1 & if m = 2 \\ 0 & otherwise, \end{cases}

the sum (46) becomes

τ((Xp_{t}X(1 − p_{t}))^{n}) = ∑_{π∈NC_{2}(2n)} ϕ_{K(π)}[p_{t}, 1 − p_{t}, …, p_{t}, 1 − p_{t}],

where NC_{2}(2n) is the set of non–crossing pairings of {1, …, 2n}. For any π ∈ NC_{2}(2n), K(π) will never have a block containing both even and odd numbers, and

ϕ_{K(π)}[p_{t}, 1 − p_{t}, …, p_{t}, 1 − p_{t}] = t^{a}(1 − t)^{b},

where a is the number of blocks of K(π) containing odd numbers and b is the number of blocks containing even numbers. Thus, if a ≥ 2, then ϕ_{K(π)}[p_{t}, 1−p_{t}, …, p_{t}, 1−p_{t}] = O(t^{2}) as t → 0. There is exactly one non–crossing pairing π of {1, …, 2n} such that K(π) groups all odd numbers into one block, namely this one:

{Xp_{t}X(1 − p_{t})Xp_{t}X(1 − p_{t})X ⋯ p_{t}X(1 − p_{t}). \atop ⋯}

This yields t in the asymptotic expansion (45) of h_{n}(t). □

Recall we let T be a DT(δ_{0}, 1)–element in a ∗–noncommutative probability space (A, τ). For n ∈ N and k_{1}, …, k_{n}, ℓ_{1}, …, ℓ_{n} ∈ N ∪ {0}, let

M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) = τ((T^{∗})^{k_{1}}T^{ℓ_{1}} … (T^{∗})^{k_{n}}T^{ℓ_{n}}).  (47)

The following properties of these moments are easily seen from Lemma 2.4 and Algorithm 2.5, and the fact (Proposition 2.12) that also T^{∗} is a DT(δ_{0}, 1)–element.

Proposition 9.4. Let n ∈ N and k_{1}, …, k_{n}, ℓ_{1}, …, ℓ_{n} ∈ N ∪ {0}.

(i) If k_{1} + ⋯ k_{n} ≠ ℓ_{1} + ⋯ + ℓ_{n} then M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) = 0.

(ii) M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) = M(ℓ_{1}, k_{2}, ℓ_{2}, …, k_{n}, ℓ_{n}, k_{1}).

(iii) M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) = M(ℓ_{n}, k_{n}, ℓ_{n−1}, k_{n−1}, …, ℓ_{1}, k_{1}).

(iv) If k_{1} = 0 and n ≥ 2 then

M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) = M(k_{2}, ℓ_{2}, …, k_{n−1}, ℓ_{n−1}, k_{n}, ℓ_{n} + ℓ_{1}).

We are now ready to state and prove the recursion formula.

Theorem 9.5. Let n ∈ N and k_{1}, …, k_{n}, ℓ_{1}, …, ℓ_{n} ∈ N, (all nonzero), be such that m := k_{1} + ⋯ + k_{n} = ℓ_{1} + ⋯ + ℓ_{n}. Then

M(k_{1},ℓ_{1}, …, k_{n}, ℓ_{n}) = \frac{1}{m + 1} ∑_{r=1}^{n} ∑_{1≤j(1)<···<j(r)≤n}  (48)

\begin{aligned}(M & (k_{1}, ℓ_{1}, …, k_{j(1)−1}, ℓ_{j(1)−1}, k_{j(1)} − 1, ℓ_{j(r)} − 1, k_{j(r)+1}, ℓ_{j(r)+1}, …, k_{n}, ℓ_{n}) \\ & ∏_{i=1}^{r−1} M(ℓ_{j(i)} − 1, k_{j(i)+1}, ℓ_{j(i)+1}, …, k_{j(i+1)−1}, ℓ_{j(i+1)−1}, k_{j(i+1)} − 1)). \end{aligned}

Proof. We assume (A, τ) = (L(F_{2}), τ) and let T = 𝒰𝒯(X, λ) with X and λ as in Lemma 4.1. We will let B denote the image of λ. For 0 < t < 1 let p_{t} = λ(1_{[0,t]}) ∈ B. Then τ(p_{t}) = t. Let S_{t} = (1 − p_{t})T and Q_{t} = p_{t}T. From Lemmas 4.6 and 4.10, we have that

(a) (1 − t)^{−1/2}S_{t} is an element of W^{∗}((1 − p_{t})X(1 − p_{t}) ∪ (1 − p_{t})B) and is a DT(δ_{0}, 1)–operator in ((1 − p_{t})A(1 − p_{t}), (1 − t)^{−1}τ↾_{(1−p_{t})A(1−p_{t})})

(b) t^{−1/2}Q_{t}p_{t} = t^{−1/2}T p_{t} is an element of W^{∗}(p_{t}Xp_{t} ∪ p_{t}B) and is a DT(δ_{0}, 1)–

operator in (p_{t}Ap_{t}, t^{−1}τ↾_{p_{t}Ap_{t}}). Substituting T = S_{t} + Q_{t} into (47) and distributing, M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) is written as a sum of the 2^{2m} terms obtained by substituting S_{t} or Q_{t} variously for each T in (47). One of these terms is

τ((S_{t}^{∗})^{k_{1}}S_{t}^{ℓ_{1}} ⋯ (S_{t}^{∗})^{k_{n}}S_{t}^{ℓ_{n}}).  (49)

Using the observation (a) about S_{t}, we see that the term (49) equals

(1 − t)^{m+1}M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}).

Thus

(1 − (1 − t)^{m+1})M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n})

is equal to the sum of the remaining 2^{2m} − 1 terms, in each of which Q_{t} or its adjoint appears at least once. As t → 0, we have the asymptotic behavior 1 − (1 − t)^{m+1} = (m + 1)t + O(t^{2}). We will show that the asympototic behavior as t → 0 of each of the remaining 2^{2m} − 1 terms is ct + O(t^{3/2}) for some constant c. The value of M(k_{1}, ℓ_{1}, …, k_{n}, ℓ_{n}) will then be the sum of all these constants, divided by m + 1.

Since p_{t}S_{t} = 0 = S_{t}p_{t}, any of these 2^{2m} − 1 terms is zero unless it is of the form

τ((S_{t}^{∗})^{k_{1}−a_{1}}(Q_{t}^{∗})^{a_{1}}Q_{t}^{b_{1}}S_{t}^{ℓ_{1}−b_{1}} ⋯ (S_{t}^{∗})^{k_{n}−a_{n}}(Q_{t}^{∗})^{a_{n}}Q_{t}^{b_{n}}S_{t}^{ℓ_{n}−b_{n}})  (50)

for some a_{j} ∈ {0, 1, …, k_{j}} and b_{j} ∈ {0, 1, …, ℓ_{j}}, and where for all j, a_{j} ≠ 0 ⟺ b_{j} ≠ 0. Writing Q_{t} = p_{t}T p_{t} + p_{t}X(1 − p_{t}) we see

τ(Q_{t}^{∗}Q_{t}) = τ(p_{t}T^{∗}p_{t}T p_{t}) + τ((1 − p_{t})Xp_{t}X(1 − p_{t})).

From the obervation (b) about p_{t}T p_{t}, we find τ(p_{t}T^{∗}p_{t}T p_{t}) = t^{2}/2. Moreover, using freeness of X and p_{t}, we find τ((1−p_{t})Xp_{t}X(1−p_{t})) = t−t^{2}, so τ(Q_{t}^{∗}Q_{t}) = t−(t^{2}/2) and ∥Q_{t}∥_{2} < t^{1/2}. On the other hand,

∥Q_{t}^{2}∥_{2} = ∥p_{t}T p_{t}T ∥_{2} ≤ ∥p_{t}T p_{t}∥ ∥Q_{t}∥_{2} ≤ 2t^{1/2}∥Q_{t}∥_{2} < 2t.

Consequently,

∥Q_{t}^{∗}Q_{t}^{2}∥_{1} ≤ ∥Q_{t}^{∗}∥_{2}∥Q_{t}^{2}∥_{2} < 2t^{3/2}.

Since ∥S_{t}∥, ∥Q_{t}∥ ≤ ∥T ∥ ≤ 2 for all t, if in (50) a_{j} ≥ 1 and b_{j} ≥ 2 for some j, then

|τ((S_{t}^{∗})^{k_{1}−a_{1}}(Q_{t}^{∗})^{a_{1}}Q_{t}^{b_{1}}S_{t}^{ℓ_{1}−b_{1}} ⋯ (S_{t}^{∗})^{k_{n}−a_{n}}(Q_{t}^{∗})^{a_{n}}Q_{t}^{b_{n}}S_{t}^{ℓ_{n}−b_{n}})| ≤

≤ ∥(S_{t}^{∗})^{k_{1}−a_{1}}(Q_{t}^{∗})^{a_{1}}Q_{t}^{b_{1}}S_{t}^{ℓ_{1}−b_{1}} ⋯ (S_{t}^{∗})^{k_{n}−a_{n}}(Q_{t}^{∗})^{a_{n}}Q_{t}^{b_{n}}S_{t}^{ℓ_{n}−b_{n}}∥_{1}

≤ 2^{2m−3}∥Q_{t}^{∗}Q_{t}^{2}∥_{1} ≤ 2^{2m−3}t^{3/2},

and similarly if a_{j} ≥ 2 and b_{j} ≥ 1. Therefore

τ((S_{t}^{∗})^{k_{1}−a_{1}}(Q_{t}^{∗})^{a_{1}}Q_{t}^{b_{1}}S_{t}^{ℓ_{1}−b_{1}} ⋯ (S_{t}^{∗})^{k_{n}−a_{n}}(Q_{t}^{∗})^{a_{n}}Q_{t}^{b_{n}}S_{t}^{ℓ_{n}−b_{n}}) = O(t^{3/2})

as t → 0, except possibly if for all j either a_{j} = 0 = b_{j} or a_{j} = 1 = b_{j}; these are the terms we shall examine in more detail. Each of them can be written in the form

τ(F_{1}(Q_{t}^{∗}Q_{t})^{d_{1}}F_{2}(Q_{t}^{∗}Q_{t})^{d_{2}} ⋯ F_{s}(Q_{t}^{∗}Q_{t})^{d_{s}})  (51)

for some s ∈ N and d_{1}, …, d_{s} ∈ N, where each F_{j} is a monomial in S_{t} and S_{t}^{∗} with F_{j} ≠ 1, or, when “s = 0,” in the form τ((Q_{t}^{∗}Q_{t})^{n}). Note that we always have F_{j} = (1 − p_{t})F_{j}(1 − p_{t}). Let us show that for all s ∈ N and d_{1}, …, d_{s} ∈ N, the asymptotic behavior

τ((1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{1}}(1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{2}} ⋯ (1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{s}}) = t + O(t^{2})  (52)

holds as t → 0. We have

Q_{t}^{∗}Q_{t} = (1 − p_{t})Xp_{t}X(1 − p_{t}) + (1 − p_{t})Xp_{t}T p_{t} + p_{t}T^{∗}p_{t}X(1 − p_{t}) + p_{t}T^{∗}p_{t}T p_{t}, so (1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{j}}(1 − p_{t}) is equal to ((1 − p_{t})Xp_{t}X(1 − p_{t}))^{d_{j}} plus some other monomials in p_{t}, (1 − p_{t}), X, T^{∗} and T, each of which contains at least one of the submonomials p_{t}T^{∗}p_{t}T p_{t} or p_{t}T p_{t}T^{∗}p_{t}. Since

∥p_{t}T^{∗}p_{t}T p_{t}∥_{1} = ∥p_{t}T p_{t}T^{∗}p_{t}∥_{1} = τ(p_{t}T^{∗}p_{t}T p_{t}) = t^{2}/2,

we obtain the asymptotic behavior

τ((1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{1}}(1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{2}} ⋯ (1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{s}})
− τ(((1 − p_{t})Xp_{t}X(1 − p_{t}))^{d_{1}+···+d_{s}}) = O(t^{2})

as t → 0. Now Lemma 9.3 shows that the asymptotic behavior (52) holds. Similar reasoning shows that for every n ∈ N, τ((Q_{t}^{∗}Q_{t})^{n}) = t + O(t^{2}) as t → 0.

Using observation (b) above we have Q_{t} ∈ W^{∗}(p_{t}X ∪ p_{t}B). Using observation (a) and Lemma 9.2, we see that the pair

{S_{t}}, {(1 − p_{t})(Q_{t}^{∗}Q_{t})^{k}(1 − p_{t}) | k ∈ N}  (53)

is ∗–free with respect to (1 − t)^{−1}τ↾_{(1−p_{t})A(1−p_{t})}. Therefore, Speicher’s moment– cummulant formula [22, Theorem 2.17] gives

τ(F_{1}(Q_{t}^{∗}Q_{t})^{d_{1}}F_{2}(Q_{t}^{∗}Q_{t})^{d_{2}} ⋯ F_{s}(Q_{t}^{∗}Q_{t})^{d_{s}}) = ∑_{π∈NC(s)}

k_{π}[F_{1}, …, F_{s}]τ_{K(π)}[(1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{1}}(1 − p_{t}), …, (1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{s}}(1 − p_{t})].

However, the asymptotic behavior (52) implies

τ_{K(π)}[(1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{1}}(1 − p_{t}), …, (1 − p_{t})(Q_{t}^{∗}Q_{t})^{d_{s}}(1 − p_{t})] =

= \begin{cases}t + O(t^{2}) & if K(π) = 1_{s}, \\ O(t^{2}) & otherwise, \end{cases}

where 1_{s} is the (trivial) partition of {1, …, s} into only one block. If K(π) = 1_{s} then π = 0_{s}, which is the partition of {1, …, s} into s blocks, and k_{0_{s}}[F_{1}, …, F_{s}] = ∏_{j=1}^{s} τ(F_{j}). Therefore,

τ(F_{1}(Q_{t}^{∗}Q_{t})^{d_{1}}F_{2}(Q_{t}^{∗}Q_{t})^{d_{2}} ⋯ F_{s}(Q_{t}^{∗}Q_{t})^{d_{s}}) = ( ∏_{j=1}^{s} τ(F_{j}))t + O(t^{2})

as t → 0.

Now let us put this together to obtain the formula (48). Of the 2^{2m} − 1 terms in which Q_{t} or its adjoint appears, after dividing by t, all contribute zero as t → 0 except those of the form

τ((S_{t}^{∗})^{k_{1}−c_{1}}(Q_{t}^{∗}Q_{t})^{c_{1}}S_{t}^{ℓ_{1}−c_{1}}(S_{t}^{∗})^{k_{2}−c_{2}}(Q_{t}^{∗}Q_{t})^{c_{2}}S_{t}^{ℓ_{2}−c_{2}} ⋯ (S_{t}^{∗})^{k_{n}−c_{n}}(Q_{t}^{∗}Q_{t})^{c_{n}}S_{t}^{ℓ_{n}−c_{n}}), where c_{1}, …, c_{n} ∈ {0, 1}. We perform the summation ∑_{r=1}^{n} ∑_{1≤j(1)<···<j(r)≤n}, let c_{j(1)} = c_{j(2)} = ⋯ = c_{j(r)} = 1 and let all other c_{i} = 0. Then the term (54) becomes

τ((S_{t}^{∗})^{k_{1}}S_{t}^{ℓ_{1}} ⋯ (S_{t}^{∗})^{k_{j(1)−1}}S_{t}^{ℓ_{j(1)−1}}(S_{t}^{∗})^{(k_{j(1)})−1}(Q_{t}^{∗}Q_{t})
S_{t}^{(ℓ_{j(1)})−1}(S_{t}^{∗})^{k_{j(1)+1}}S_{t}^{ℓ_{j(1)+1}} ⋯ (S_{t}^{∗})^{k_{j(2)−1}}S_{t}^{ℓ_{j(2)−1}}(S_{t}^{∗})^{(k_{j(2)})−1}(Q_{t}^{∗}Q_{t})
⋮
S_{t}^{(ℓ_{j(r−1)})−1}(S_{t}^{∗})^{k_{j(r−1)+1}}S_{t}^{ℓ_{j(r−1)+1}} ⋯ (S_{t}^{∗})^{k_{j(r)−1}}S_{t}^{ℓ_{j(r)−1}}(S_{t}^{∗})^{(k_{j(r)})−1}(Q_{t}^{∗}Q_{t})
S_{t}^{(ℓ_{j(r)})−1}(S_{t}^{∗})^{k_{j(r)+1}}S_{t}^{ℓ_{j(r)+1}} ⋯ (S_{t}^{∗})^{k_{n}}S_{t}^{ℓ_{n}}) =
= τ( \begin{aligned}S_{t}^{(ℓ_{j(r)})−1}(S_{t}^{∗})^{k_{j(r)+1}}S_{t}^{ℓ_{j(r)+1}} ⋯ (S_{t}^{∗})^{k_{n}}S_{t}^{ℓ_{n}} & (S_{t}^{∗})^{k_{1}}S_{t}^{ℓ_{1}} ⋯ \\ & ⋯ (S_{t}^{∗})^{k_{j(1)−1}}S_{t}^{ℓ_{j(1)−1}}(S_{t}^{∗})^{(k_{j(1)})−1}(Q_{t}^{∗}Q_{t}) \end{aligned}

S_{t}^{(ℓ_{j(1)})−1}(S_{t}^{∗})^{k_{j(1)+1}}S_{t}^{ℓ_{j(1)+1}} ⋯ (S_{t}^{∗})^{k_{j(2)−1}}S_{t}^{ℓ_{j(2)−1}}(S_{t}^{∗})^{(k_{j(2)})−1}(Q_{t}^{∗}Q_{t})
⋮
S_{t}^{(ℓ_{j(r−1)})−1}(S_{t}^{∗})^{k_{j(r−1)+1}}S_{t}^{ℓ_{j(r−1)+1}} ⋯ (S_{t}^{∗})^{k_{j(r)−1}}S_{t}^{ℓ_{j(r)−1}}(S_{t}^{∗})^{(k_{j(r)})−1}(Q_{t}^{∗}Q_{t})).

Divided by (m + 1)t, this quantity tends in the limit as t → 0 to

\frac{1}{m + 1}M(ℓ_{j(r)} − 1, k_{j(r)+1}, ℓ_{j(r)+1}, …, k_{n}, ℓ_{n}, k_{1}, ℓ_{1}, …, k_{j(1)−1}, ℓ_{j(1)−1}, k_{j(1)} − 1)·

· ∏_{i=1}^{r−1} M(ℓ_{j(i)} − 1, k_{j(i)+1}, ℓ_{j(i)+1}, …, k_{j(i+1)−1}ℓ_{j(i+1)−1}, k_{j(i+1)} − 1)

= \frac{1}{m + 1}M(k_{1}, ℓ_{1}, …, k_{j(1)−1}, ℓ_{j(1)−1}, k_{j(1)} − 1, ℓ_{j(r)} − 1, k_{j(r)+1}, ℓ_{j(r)+1}, …, k_{n}, ℓ_{n})·

· ∏_{i=1}^{r−1} M(ℓ_{j(i)} − 1, k_{j(i)+1}, ℓ_{j(i)+1}, …, k_{j(i+1)−1}ℓ_{j(i+1)−1}, k_{j(i+1)} − 1) .

Thus we obtain (48). □

We finish this section and the paper with a conjecture.

Conjecture 9.6. For all k, n ∈ N,

τ(((T^{∗})^{k}T^{k})^{n}) = \frac{n^{nk}}{(nk + 1)!}.

The conjecture was proved in the case k = 1 in Theorem 8.7. It is easy to prove in the case n = 1, either from the recursion formula (48) or using Lemma 2.4, and either of these techniques with a little more work can be used to prove it in the case n = 2, (or see the proof by contour integration of a generating function in [7]). We have, without making an effort to go very far, checked some additional instances of the conjecture on a computer using the recursion formula (48). The following table lists the cases of the conjecture that we have either proved or checked by computation:

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 all n
all k all k k ≤ 60 k ≤ 25 k ≤ 15 k ≤ 10 k ≤ 5 k = 1.

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Ken Dykema, Department of Mathamtics, Texas A&M University, College Station TX 77843–3368, USA

  E-mail address: Ken.Dykema@math.tamu.edu

Uffe Haagerup, Department of Mathematics and Computer Science, University of Southern Denmark — Odense University, Campusvej 55, 5230 Odense M, Denmark

  E-mail address: haagerup@imada.sdu.dk