FUNCTIONS WHICH ARE ALMOST MULTIPLIERS OF HILBERT FUNCTION SPACES

J. AGLER AND N. J. YOUNG

Abstract. We introduce a natural class of functions, the pseudomultipliers, associated with a general Hilbert function space, prove an extension theorem which justifies the definition, give numerous examples and establish the nature of the 1-pseudomultipliers of Hilbert spaces of analytic functions under mild hypotheses.

The function 1/z on the unit disc 𝔻 is almost a multiplier of the Hardy space H^{2}: it misses by only one dimension. That is, there is a closed subspace of H^{2} of codimension 1 which is multiplied by 1/z into H^{2}. The same statement holds for the characteristic function of the point 0. These are two key examples of functions that we call pseudomultipliers of Hilbert function spaces. Now multipliers of the standard function spaces have been much studied: it is a natural generalization to consider functions which fail to be multipliers by only finitely many dimensions. Moreover, for some familiar function spaces one obtains in this way natural classes of functions. For example, a well-known theorem of Adamyan, Arov and Krein [AAK] on s-numbers of Hankel operators can be interpreted as a description of the pseudomultipliers of H^{2} (see Theorem 3.4 below). They are the finite modifications of functions of the form f/p where f is bounded and analytic in the unit disc and p is a polynomial which does not vanish on the unit circle. The pseudomultipliers of the Fock space (see Theorem 3.5 below) are the finite modifications of the proper rational functions. We address the question of what can be said about the pseudomultipliers of other popular function spaces.

One can formulate the definition of a pseudomultiplier of a function space in very great generality. In this paper we introduce the notion for the case of Hilbert function spaces. This still covers a very wide variety of spaces, but we are nevertheless able to make significant assertions about them. A major purpose of the paper is to give the “correct” formulation of the notion of a pseudomultiplier. To justify our definition we establish an extension theorem (Theorem 2.1). This is the main result of the paper and shows that apparently weaker variants of our definition give rise to essentially the same objects. Other goals are to illustrate the notion by means of a range of examples and to show that a surprising amount can be said about pseudomultipliers of Hilbert spaces of analytic functions under modest hypotheses. Virtually all our results depend heavily on a key technical fact, Lemma 2.5.

Date: May 23, 1996.

1991 Mathematics Subject Classification. AMS Subject Classifications: 46E22, 46E20.

Key words and phrases. reproducing kernel, multiplier, Pick’s theorem, Adamyan-Arov-Krein theorem.

Agler’s research was supported by an NSF grant in Modern Analysis. Young wishes to thank the University of California at San Diego and the Mathematical Sciences Research Institute for hospitality while this work was carried out. Research at MSRI is supported in part by NSF grant DMS-9022140.

Our investigation began as a study of certain interpolation problems. It is known [Ag1, Ag2, Q1] that the classical result of Pick on interpolation by bounded analytic functions [P] can be extended to certain other function spaces, some of them having no connection with analyticity. Could the same be true of the interpolation theorem of Adamyan, Arov and Krein which generalises Pick’s theorem? This question was discussed but not resolved in [Q2]. To answer it we had to analyse the “Pick kernel”

(λ, μ) ↦ (1 − φ(λ)φ̄(μ))k(λ, μ) (0.1)

corresponding to a reproducing kernel k and a function φ. Pick’s theorem concerns the positivity of this kernel on 𝔻 when k is the Szegő kernel, while the result of Adamyan, Arov and Krein tells us what happens when it has m negative squares on 𝔻. A study of the same question for other kernels led us to the notion of a pseudomultiplier, which proved to be a remarkably fruitful notion despite its great generality. We think of pseudomultipliers as being something like meromorphic functions (plus point discontinuities), though they are defined on arbitrary sets which need have no differentiable or even topological structure - all information is contained in the kernel. Regarding the original question, we shall show in a future paper that, for the most straightforward formulation, the answer is negative: the Adamyan-Arov-Krein property characterizes the Szegő kernel. However, pseudomultipliers are interesting on their own account.

In this paper a kernel on Ω will mean a function

k : Ω × Ω → ℂ,

where Ω is a set, satisfying

k(λ, μ) = k(μ, λ)^{−} for all λ, μ ∈ Ω.

We say that k is nonsingular if, for any n ∈ ℕ and λ_{1}, …, λ_{n} ∈ Ω, the n × n matrix [k(λ_{i}, λ_{j})]_{i,j=1}^{n} is nonsingular. We call k a positive kernel if every [k(λ_{i}, λ_{j})] above is positive, and positive definite if it is both positive and nonsingular. For any nonnegative integer m, we say that a kernel k has m negative squares if, for any n ∈ ℕ and λ_{1}, …, λ_{n} ∈ Ω, the n × n matrix [k(λ_{i}, λ_{j})]_{i,j=1}^{n} has no more than m negative eigenvalues (counted with multiplicity), while for some choice of n, λ_{1}, …, λ_{n}, it has exactly m negative eigenvalues.

If k is a positive kernel on a set Ω, there is an associated Hilbert space H(k) of functions on Ω for which k is the reproducing kernel [Ar]. That is, for any f ∈ H(k) and λ ∈ Ω, we have

⟨f, k_{λ}⟩_{H(k)} = f(λ),

where k_{λ} ∈ H(k) is defined by k_{λ}(μ) = k(μ, λ) for μ ∈ Ω. By a Hilbert space of functions on a set Ω we shall mean a Hilbert space H whose elements are functions on Ω such that the point evaluations h ↦ h(λ) are continuous linear functionals on H for all λ ∈ Ω. Such a space has a reproducing kernel which is a positive kernel on Ω; it may fail to be nonsingular. If H is a Hilbert space of functions on Ω with kernel k and E is a subset of Ω then the space of restrictions of elements of H to E is a Hilbert space of functions on E, whose kernel is the restriction of k to E. If E is large enough these two Hilbert spaces are naturally isomorphic, but for our purposes they are to be regarded as two distinct Hilbert function spaces. This point will be important when we discuss domains of definition of functions.

We say that a function ψ : Ω → ℂ is a multiplier of H(k) if ψf ∈ H(k) for every f ∈ H(k). When ψ is a multiplier of H(k), the mapping f ↦ ψf is a linear operator, which we denote by M_{ψ}. By the closed graph theorem, M_{ψ} is bounded. The multiplier norm of ψ is ∥M_{ψ}∥. The set of all multipliers of H(k) is a subalgebra of the algebra ℒ(H(k)) of all bounded linear operators on H(k). It is well-known [BB, S] (and will follow from Theorem 2.1 below) that, for any function ψ : Ω → ℂ, the following two statements are equivalent:

(a) ψ is a multiplier of H(k) and ∥M_{ψ}∥ ≤ 1, and

(b) the kernel (λ, μ) ↦ (1 − ψ(λ)ψ̄(μ))k(λ, μ) is positive on Ω.

If H is a Hilbert space and u, v ∈ H we denote by u ⊗ v the rank 1 operator on H given by x ↦ ⟨x, v⟩u. If 𝒞 is a class of functions on a set Ω we say that E ⊂ Ω is a set of uniqueness for 𝒞 if f, g ∈ 𝒞 and f|E = g|E imply f = g (here f|E is the restriction of f to E). If k is a positive definite kernel on Ω then E ⊂ Ω is a set of uniqueness for H(k) if and only if {k_{λ} : λ ∈ E} spans a dense linear subspace of H(k). For a space H of functions on Ω we shall have occasion to use the class

HH + HH =^{def} {f_{1}f_{2} + f_{3}f_{4} : f_{j} ∈ H for 1 ≤ j ≤ 4}.

We denote by s_{j}(T), for j ≥ 0, the s-numbers of a bounded linear operator T on a Hilbert space H:

s_{j}(T) = inf{∥T − R∥ : R ∈ ℒ(H), rank R ≤ j}.

We remark that nonsingularity of a positive kernel k on a set Ω can be expressed in terms of its associated Hilbert space H of functions. It is easy to show that k is nonsingular on Ω if and only if, for every finite sequence λ_{1}, …, λ_{n} of points of Ω, there is a function h ∈ H such that h(λ_{1}) = 1 and h(λ_{j}) = 0 for 2 ≤ j ≤ n. Another equivalent condition is that {k_{λ} : λ ∈ Ω} is a linearly independent set in H.

1. Pseudomultipliers of Hilbert Function Spaces

The starting point for the notion of an m-pseudomultiplier of a Hilbert space H of functions on a set Ω is simply a function ψ on Ω such that, for some closed mcodimensional subspace E of H, ψE ⊂ H. However, as our archetype 1/z shows, we need to allow the possibility that ψ is not defined on the whole of Ω, and this gives rise to some delicate issues. We illustrate these issues by examples in order to motivate the formal definition. In the two examples which follow k is the Szegő kernel and Ω is 𝔻, so that H is H^{2}. We consider functions ψ : D_{ψ} ⊂ Ω → ℂ.

Example 1.1. D_{ψ} = 𝔻 \ {0} and ψ(λ) = λ^{−m}, where m ∈ ℕ. Here z^{m}H^{2} has codimension m in H^{2} and ψ multiplies z^{m}H^{2} into H^{2}. Multiplication by ψ determines a bounded linear operator of norm 1 from z^{m}H^{2} into H^{2}; we say that ψ is an mpseudomultiplier of H^{2} and that s_{m}(ψ) = 1.

Example 1.2. D_{ψ} = 𝔻 and ψ is given by ψ(0) = 1 and ψ(λ) = 0 if λ ≠ 0. Here multiplication by ψ determines the zero operator from the 1-codimensional subspace zH^{2} of H^{2} into H^{2}. Thus ψ is a 1-pseudomultiplier and s_{1}(ψ) = 0 . However ψ is not a 0-pseudomultiplier.

What should we assume about the domain of an m-pseudomultiplier? It should be in some sense “almost all” of Ω, but in what sense? There are various choices one could make; an important purpose of this paper is to show that all the natural assumptions about the domain lead to essentially the same objects, and these are functions which are defined everywhere on Ω with the exception of at most m points. Accordingly we shall incorporate this requirement into the definition, and in the following section we shall justify our decision by showing that weaker assumptions lead to the same class of functions.

A 1-pseudomultiplier of H^{2}, then, will be a function defined on the complement of at most one point in 𝔻. Consider the function ψ which is identically zero on 𝔻 \ {0}. This ψ satisfies ψzH^{2} = {0} ⊂ H^{2}. Should we regard ψ as a pseudomultiplier? And as distinct from the function which is identically zero on all of 𝔻? We prefer to regard these two functions as the same, and so will include in our definition a condition to ensure that if a pseudomultiplier is undefined at a point of Ω then it is in a precise sense not definable at that point.

Now consider a fixed Hilbert space H of functions on a set Ω with nonsingular kernel k. For D ⊂ Ω and any function f on Ω we denote by f|D the restriction of f to D, and we denote by H|D the set {h|D : h ∈ H}. For ψ : D_{ψ} ⊂ Ω → ℂ and h ∈ H we denote by ψh the pointwise product of ψ and h|D_{ψ}, so that ψh is a function on D_{ψ}. For E ⊂ H we denote by ψE the set {ψh : h ∈ E}, so that ψE is a set of functions on D_{ψ}. If E is a closed linear subspace of H, if D_{ψ} is a set of uniqueness and ψE ⊂ H|D_{ψ} we define the linear operator M_{ψ,E} : E → H by M_{ψ,E}h = g where g is the unique element of H which extends ψh. By the closed graph theorem M_{ψ,E} is continuous. If D_{ψ} is a set of uniqueness of H we shall say that α ∈ Ω is a singularity of ψ (relative to H) if there exists h ∈ H such that h(α) = 0 and ψh has an extension to a function g ∈ H (necessarily unique) such that g(α) ≠ 0. Of course this can only happen if α ∉ D_{ψ}.

Definition 1.3. Let ψ be a complex-valued function on a subset D_{ψ} of Ω and let m ∈ ℤ^{+}, the set of non-negative integers. We say that ψ is an m-pseudomultiplier of H on Ω if

1. D_{ψ} is a set of uniqueness for H;

2. Ω \ D_{ψ} consists of singularities of ψ and contains at most m points;

3. there is a closed subspace E of codimension m in H such that ψE ⊂ H|D_{ψ}. For an m-pseudomultiplier ψ of H we define s_{m}(ψ) to be the infimum of ∥M_{ψ,E}∥

over all closed m-codimensional subspaces E of H such that ψE ⊂ H|D_{ψ}. We say that ψ is a pseudomultiplier of H if it is an m-pseudomultiplier of H for some nonnegative integer m.

Notes 1. In the case of a finite set Ω no proper subset is a set of uniqueness (recall that we have supposed the kernel k nonsingular), and so pseudomultipliers are everywhere defined on Ω. Every ψ : Ω → ℂ is an m-pseudomultiplier and s_{m}(ψ) is the mth singular value of the multiplication operator M_{ψ}.

2. A 0-pseudomultiplier ψ is the same as a multiplier of H, and s_{0}(ψ) = s_{0}(M_{ψ}).

3. If m ≥ 1, ψ is defined at all but m−1 points of Ω and ψ is an m-pseudomultiplier but not an (m − 1)-pseudomultiplier then there is a unique E with the stated properties, and so s_{m}(ψ) = ∥M_{ψ,E}∥. For if E_{1}, E_{2} are distinct and both have all the stated properties then E_{1} + E_{2} is a closed subspace of H of codimension at most m − 1 and ψ(E_{1} + E_{2}) ⊂ H|D_{ψ}. That is, ψ is an (m − 1)-pseudomultiplier.

In particular, if a 1-pseudomultiplier ψ is defined on the whole of Ω then unless it is a multiplier, there is a unique 1-codimensional subspace E of H such that ψE ⊂ H. However, it can happen for a 1-pseudomultiplier ψ for which D_{ψ} ≠ Ω that ψH ⊂ H|D_{ψ} and yet ψ cannot be extended to a multiplier of H: see Example 2.2 below.

4. A related definition of s_{m}(ψ) and an analysis in the case of “Blaschke kernels” k is in [Q2].

5. For ψ as in the Definition, Ω \ D_{ψ} consists precisely of the singularities of ψ. The notion of a pseudomultiplier has as much generality as that of a Hilbert func-

tion space, and so it is not to be expected that we can give a very precise taxonomy of pseudomultipliers in general. We do want to give a wide range of examples to illustrate the notion; to introduce a little order into these examples we shall take a preliminary step towards a classification of 1-pseudomultipliers. For this purpose we shall make an additional (but mild) assumption on the pseudomultiplier ψ : D_{ψ} → ℂ, to wit that D_{ψ} is a set of uniqueness for the class HH + HH of functions on Ω. A simple distinction is between the ψ for which D_{ψ} = Ω and those for which Ω \ D_{ψ} contains exactly one point. We shall say that ψ has defect 0 or 1 according as the first or second of these holds. Consider first the case of defect 1. There exists a unique α ∈ Ω \ D_{ψ}, and there are u, g_{1} ∈ H such that u(α) = 0, ψu = g_{1}|D_{ψ}, and g_{1}(α) ≠ 0. There is also a closed 1-codimensional subspace E of H such that ψE ⊂ H|D_{ψ}. We claim that

E = k_{α}^{⊥} = {h ∈ H : h(α) = 0}.

For suppose that h ∈ H is such that ψh ∈ H|D_{ψ} – say ψh = g_{2}|D_{ψ}, g_{2} ∈ H. On D_{ψ} we have

g_{1}h − ug_{2} = ψuh − ψuh = 0,

and so, by the hypothesis on HH + HH, g_{1}h = ug_{2} on Ω. Since u(α) = 0 and g_{1}(α) ≠ 0 we have h(α) = 0. Thus E ⊂ k_{α}^{⊥}. Since E has codimension 1 it must be that E = k_{α}^{⊥}. We have shown that, subject to the assumption on HH + HH, if ψ is a 1-pseudomultiplier of defect 1 and D_{ψ} = Ω \ {α}, then ψk_{α}^{⊥} ⊂ H|D_{ψ}, and moreover E = k_{α}^{⊥} is the only closed 1-codimensional subspace of H such that ψE ⊂ H|D_{ψ}.

Now suppose that ψ is of defect 0, i.e. ψ is everywhere defined on Ω. We identify three subcases.

1. ψ is a multiplier of H.

2. ψ is not a multiplier and there is a point α ∈ Ω such that ψk_{α}^{⊥} ⊂ k_{α}^{⊥}.

3. ψ is not a multiplier and there is a closed subspace E of codimension 1 in H such that ψE ⊂ H|D_{ψ} and E has no zeros in Ω.

These three cases do exhaust the 1-pseudomultipliers of defect 0. For if ψ is not a multiplier then, by Note 3 above, there is a unique closed 1-codimensional subspace E of H such that ψE ⊂ H|D_{ψ}. If E is of the form k_{α}^{⊥} for some α ∈ Ω then we must have ψk_{α}^{⊥} ⊂ k_{α}^{⊥} since ψ(α) is defined, and we are in case 2. The remaining possibility is that E is not k_{α}^{⊥} for any α, and case 3 applies.

Let us exhibit examples of all these cases. Defect 1 is exemplified by Example 1.1 (ψ(z) = 1/z). Defect 0, case 1 is multipliers – constant functions will do. Example 1.2 (the characteristic function of a point) illustrates defect 0, case 2. Defect 0, case 3 occurs in the following.

Example 1.4. Let k be the Szegő kernel restricted to the disc Ω = \{λ ∈ ℂ : |λ− \frac{1}{2}| < \frac{1}{2}\}, take D_{φ} = Ω and φ(λ) = λ^{−1}. Then s_{1}(φ) = 1. As a Hilbert space H is the usual Hardy space H^{2}, but as a function space it is a space of functions on Ω. Multiplication by φ acts contractively from 1^{⊥} into H, but the elements of 1^{⊥} have no common zero in Ω.

The latter example looks somewhat unnatural, because we took k on the “wrong” domain Ω, but we are working in such generality that such displeasing examples are included. In the next section we shall make further assumptions on k in order to exclude such phenomena, and we shall be able to make stronger assertions about the structure of pseudomultipliers. Let us return to defect 0, case 2: ψk_{α}^{⊥} ⊂ k_{α}^{⊥}. Note that ψ is defined at α, but that the value of s_{1}(ψ) is independent of ψ(α). We could redefine ψ(α) without affecting the fact that ψ is a 1-pseudomultiplier or the value of s_{1}(ψ). We shall call such a phenomenon an anomaly of ψ. Our example 1.2, the characteristic function of the point 0, has an anomaly at 0. In this example it is possible to redefine ψ(0) to obtain a multiplier of H. This is one easy way to generate 1-pseudomultipliers of defect 0, case 2: we will always obtain either such an object or a multiplier if we take a multiplier and change its value at a single point. However, not all instances of defect 0, case 2 arise in this way.

Example 1.5. Let Ω = 𝔻 and let H = zH^{2} ⊕ ℂg where g(z) = (z − 1)^{−1} and the norm in ℂg is given by ∥λg∥ = |λ|. Let ψ(z) = z if z ≠ 0 and let ψ(0) = c. Then ψH(0) ⊂ H(0), so that 0 is an anomaly of ψ, but there is no choice of c which makes ψ a multiplier of H. For suppose that ψg ∈ H. Then

ψg = zf + λg for some f ∈ H^{2}, λ ∈ ℂ.

On putting z = 0 we find that λ = c, and so

f(z) = \frac{ψ(z) − c}{z}g(z) = \frac{z − c}{z(z − 1)} for z ≠ 0.

This contradicts f ∈ H^{2}, whatever the choice of c ∈ ℂ.

Let us extend the terminology introduced above. For a closed subspace F of H we denote by P_{F} the orthogonal projection operator from H to F.

Definition 1.6. Let ψ be a m-pseudomultiplier of H. We say that ψ is of defect j if the complement of the domain of ψ contains j points. We say that α ∈ Ω is an anomalous point or an anomaly of ψ if ψk_{α}^{⊥} ⊂ k_{α}^{⊥}.

Here are further examples of the variety of behaviour of pseudomultipliers.

Example 1.7. A pole which is not a singularity. H = H^{2}, D_{φ} = 𝔻 and

φ(λ) = \begin{cases}λ^{−m} & if λ ≠ 0, \\ c & if λ = 0, \end{cases}

where m ∈ ℕ and c ∈ ℂ. This example differs from Example 1.1 in that φ is defined at 0. Here φ multiplies z^{m+1}H, but no m-codimensional subspace of H, into H. It is thus an (m + 1)-pseudomultiplier but not an m-pseudomultiplier. It has defect 0.

In connection with examples 1.1 and 1.7 we pose the question: is there a natural notion of the multiplicity of a singularity of a pseudomultiplier for a general kernel?

Example 1.8. The simplest of all infinite-dimensional Hilbert function spaces is ℓ^{2} as a space of functions on ℕ in the natural way. Its kernel is k(λ, μ) = δ_{λμ}, the Kronecker symbol. No proper subset of ℕ is a set of uniqueness for ℓ^{2}, and so pseudomultipliers are defined on all of ℕ. In fact the pseudomultiplier algebra of ℓ^{2} is ℓ^{∞}, which is also the multiplier algebra. For suppose there is an unbounded m-pseudomultiplier ψ. We may pick an increasing sequence (n_{j}) in ℕ such that |ψ(n_{j})| ≥ j and a space E of codimension m in ℓ^{2} such that ψE ⊂ ℓ^{2}. Let P_{N} be the orthogonal projection operator from ℓ^{2} to span{k_{1}, k_{2}, …, k_{N}}. Choose N so large that P_{N}E^{⊥} has dimension m. Then, if f_{1}, …, f_{m} is a basis of E^{⊥}, the vectors P_{N}f_{1}, …, P_{N}f_{m} are linearly independent. For n > N define x(n) to be 1/j if n = n_{j}, 0 otherwise. Pick ξ ∈ span{k_{1}, …, k_{N}} such that

< ξ, P_{N}f_{j} >= − ∑_{n=N+1}^{∞} x(n)f̄_{j}(n), for 1 ≤ j ≤ m.

This is possible on account of the linear independence of the P_{N}f_{j}. Now set x(n) = ξ(n) for 1 ≤ n ≤ N: then x⊥f_{j} for 1 ≤ j ≤ m. Thus x ∈ E and hence ψx ∈ ℓ^{2}. However |ψx(n_{j})| ≥ 1 for infinitely many points n_{j}, a contradiction. Thus m-pseudomultipliers of ℓ^{2} are bounded.

Example 1.9 (The Fock space Φ). This is the space with kernel k(λ, μ) = e^{λμ̄} on ℂ. It is also known as the Fischer space and the Bargmann–Fock space. It comprises all entire functions f such that

∫_{ℂ} |f(z)|^{2}e^{−|z|^{2}} dA < ∞

where dA is area measure. Alternatively it can be described as the space of functions f(z) = ∑_{n=0}^{∞} a_{n}z^{n} such that ∑_{n=0}^{∞} |a_{n}|^{2}n! < ∞. An orthonormal basis of Φ is given by the functions (z^{n}/\sqrt{n!} )_{n=0}^{∞}. The interest of Φ here is that it admits no nonconstant multipliers, but it has plenty of nontrivial pseudomultipliers. Indeed, a non-trivial Hilbert space of entire functions cannot have a non-constant multiplier. For suppose that φ is such a multiplier. It is easily shown that φ is entire, and so, by Liouville’s theorem, φ(ℂ) is unbounded. Since H is non-trivial its kernel k satisfies k_{λ} ≠ 0 for λ in a dense subset of ℂ. The relation M_{φ}^{∗}k_{λ} = φ̄(λ)k_{λ} shows that φ(ℂ) ⊂ σ(M_{φ}), contradicting boundedness of M_{φ}. In particular therefore the only multipliers of the Fock space are the constants, but φ : ℂ \ {0} → ℂ with φ(λ) = λ^{−1} is a 1pseudomultiplier: it multiplies the 1-codimensional subspace {h ∈ H : h(0) = 0} of H into H. We shall discuss the pseudomultipliers of Φ further in Sec. 3 below.

H^{2} has the property that if ψ is a multiplier then s_{m}(ψ) = s_{0}(ψ) for all m ∈ ℤ^{+}. A simple example of a space which does not have this property is ℓ^{2}, as a space of functions on ℕ. Here ψ(n) = 1/n defines a multiplier of ℓ^{2}; clearly s_{m}(ψ) = 1/(m+1).

One might imagine that a meromorphic function defining a 1-pseudomultiplier would have a single, simple pole. This is true for nonsingular kernels, as will follow from Theorem 2.6 below, but it is not true if we relax for a moment our standing assumption that H has nonsingular kernel, as the following example shows.

Example 1.10. Let m ≥ 2 and let H be the closure in H^{2} of the space of polynomials in λ^{m}, as a space of functions on 𝔻, and let φ(λ) = λ^{−m} on 𝔻\{0}. Then φ multiplies 1^{⊥} into H, hence is a 1-pseudomultiplier, but has a pole of multiplicity m at 0.

Examples 1.1 and 1.9 suggest a conjecture: if H is a Hilbert space of analytic functions on a domain Ω with nonsingular kernel, then for any multiplier ψ of H and any α ∈ Ω then the function λ ↦ ψ(λ)/(λ − α) is a 1-pseudomultiplier of H. However, this is not so, even if H is z-invariant.

Example 1.11. Let Ω = 𝔻 and D_{φ} = 𝔻 \ {0}, φ(λ) = λ^{−1}. Let H be a z-invariant subspace of the Bergman space L_{a}^{2}(𝔻) with nonsingular kernel such that zH has codimension 2 in H. Here z denotes the operation of multiplication by the independent variable. It is shown in [H] how such a subspace can be constructed. One can take H to be of the form I(A)+I(B) where A, B are certain zero sequences of L_{a}^{2}(𝔻) and the closed subspace I(A) is {f ∈ L_{a}^{2}(𝔻) : f = 0 on A}. It is shown that A, B can be chosen so that A∪B is a “sampling sequence” for L_{a}^{2}(𝔻), which ensures that I(A)+I(B) is closed. If λ_{1}, …, λ_{n} are distinct points of 𝔻 one can find f ∈ I(A)+I(B) such that f(λ_{1}) ≠ 0; then the function ∏_{j=2}^{n}(λ − λ_{j})f(λ) ∈ H vanishes at λ_{2}, …, λ_{n} but not at λ_{1} and so H has nonsingular kernel. Now suppose that φ is a 1-pseudomultiplier of H. By Definition 1.3 there is a subspace E of codimension one in H and a bounded linear operator L : E → H such that (Lh)(λ) = λ^{−1}h(λ) for all h ∈ E and all but one λ ∈ 𝔻 \ {0}. Thus each h ∈ E agrees with an element of zH except for at most two points, and so belongs to zH. Thus E ⊂ zH, a contradiction since E has codimension one and zH has codimension two. Thus φ is not a 1-pseudomultiplier.

By a slight modification of the above construction one can find a z-invariant H ⊂ L_{a}^{2}(𝔻) with nonsingular kernel such that φ(λ) = λ^{−1} is not an m-pseudomultiplier for any integer m. It is shown in [HRS] that H can be chosen so that zH has infinite codimension in H. It follows that φ is not a pseudomultiplier.

Most of our natural-looking examples of pseudomultipliers relate to Hilbert spaces of analytic functions, so it is of interest to show that other types of space can also have non-trivial pseudomultipliers.

Example 1.12. A Sobolev space. Let W be the space {f ∈ L^{2}(0, 1) : f′ ∈ L^{2}(0, 1)}, with its usual inner product, as a space of functions on [0, 1]. The characteristic function of a point is a pseudomultiplier of this space, but a much more interesting example is the function φ(t) = \sqrt{t} . Since W is an algebra with identity and φ ∉ W it follows that φ is not a multiplier of W, but φ is a 1-pseudomultiplier of W. In fact φk_{0}^{⊥} ⊂ W. Consider f ∈ W such that f(0) = 0. Then for t ∈ (0, 1)

|f(t)| = |∫_{0}^{t} f′(x)dx| ≤ \sqrt{t}||f||_{W} ,

so that φ′f is bounded on (0, 1). It follows that φf ∈ W. Hence φk_{0}^{⊥} ⊂ W as claimed.

We shall need the following simple fact.

Theorem 1.13. If φ is a pseudomultiplier with domain D_{φ} and φ is the restriction of a multiplier of H then D_{φ} = Ω and φ is a multiplier of H.

Proof. We have to show that φ cannot have any singularities in Ω. Suppose, on the contrary, that φ is the restriction of a multiplier θ and that there exist α ∈ Ω and h ∈ H such that h(α) = 0 and φh has an extension g ∈ H such that g(α) ≠ 0. Now θh ∈ H extends φh, and so θh = g.Thus

0 = θ(α)h(α) = g(α) ≠ 0,

a contradiction.

2. Extension of pseudomultipliers and the Pick kernel

In our definition of an m-pseudomultiplier φ of a Hilbert space H of functions on a set Ω the main idea is that φ multiply a closed m-codimensional subspace of H into H. This is natural enough, as is the requirement that the domain D_{φ} of φ be a set of uniqueness for H; otherwise φ is more appropriately studied via its action on the proper closed subspace of H spanned by k_{λ}, λ ∈ D_{φ}. However, we have also imposed the condition 2 of Definition 1.3, that Ω \ D_{φ} consist of singularities of φ and have at most m points. This is less obviously appropriate. Why make such a strong assumption on D_{φ}? It would make perfect sense simply to delete condition 2. Let us temporarily call functions satisfying the resulting definition weak m-pseudomultipliers. On the face of it these will comprise a larger class than the mpseudomultipliers. It is a remarkable fact that under a modest hypothesis on D_{φ} every weak m-pseudomultiplier can be extended with preservation of s_{m}(·) to a function which is defined on all but at most m points of Ω and which is an m-pseudomultiplier.

Observe that if D_{φ} is a set of uniqueness for H then H|D_{φ} =^{def} {h|D_{φ} : h ∈ H} is a space of functions on D_{φ}, and h ↦ h|D_{φ} is a Hilbert space isomorphism of H and H|D_{φ}. To say that φ is a weak m-pseudomultiplier of H is the same as saying that φ is an m-pseudomultiplier of H|D_{φ}. Our main aim in this section is to show that an m-pseudomultiplier of H|D_{φ} extends with preservation of s_{m}(·) to an mpseudomultiplier of H. In fact this is not true in full generality, even for multipliers (m = 0), as we show in Example 2.2 below, but the extra hypothesis we need to obtain such an extension holds for nearly all spaces of interest.

Theorem 2.1. Let H be a Hilbert space of functions on a set Ω with nonsingular kernel k, let D_{φ} ⊂ Ω be a set of uniqueness for H and for HH + HH. The following are equivalent for a function φ : D_{φ} → ℂ.

1. φ is an m-pseudomultiplier of H|D_{φ};

2. There exists t ≥ 0 such that the kernel

(λ, μ) ↦ (t^{2} − φ(λ)φ̄(μ)) k(λ, μ) (2.1)

has at most m negative squares on D_{φ};

3. There is an m-pseudomultiplier ψ of H which extends φ. Moreover, when these conditions hold,

s_{m}(φ) = s_{m}(ψ) = inf t

over all t for which (2.1) has at most m negative squares.

Before we prove the theorem let us look at an example which shows how the conclusions can fail if the hypothesis about HH + HH is omitted.

Example 2.2. Let Ω = ℕ, let u = \sqrt{3}(2^{−n})_{n=1}^{∞}, so that u is a unit vector in ℓ^{2}, and let H be the orthogonal complement of u in ℓ^{2}. The reproducing kernel of ℓ^{2} is (λ, μ) ↦ δ_{λμ} and so the reproducing kernel in H is given by orthogonal projection onto H:

k_{λ}(μ) = [δ_{λ} − ⟨δ_{λ}, u⟩u](μ) = δ_{λμ} − \frac{3}{2^{λ+μ}} .

This example is simple, but it has some interesting properties. It illustrates pathologies which can occur when pseudomultipliers are studied in excessive generality.

(a) Only the constant sequences are multipliers of H, but there is a rich class of pseudomultipliers. In fact ψ is a pseudomultiplier of H ⇔ ψ is a 1- pseudomultiplier of H ⇔ either ψ ∈ ℓ^{∞} or ψ is a bounded non-constant function on ℕ \ {α} for some α ∈ ℕ.

(b) For any α ∈ ℕ and any non-constant bounded function φ : ℕ \ {α} → ℂ, φ is a multiplier of H|ℕ \ {α} but φ has no extension to a multiplier of H.

(c) For φ as in (b) and t = sup{|φ_{n}| : n ∈ ℕ \ {α}}, the Pick kernel

(λ, μ) ↦ (t^{2} − φ(λ)φ̄(μ)) k(λ, μ)

is positive but φ does not extend to a multiplier of H.

(d) There are functions ψ such that ψH ⊂ H|D_{ψ} but ψ is not a multiplier of H, nor even extendable to a multiplier of H.

Notes. Since ℕ \ {α} is a set of uniqueness for H, for any α ∈ ℕ, (b) shows that the implication 1 ⇒ 3 of Theorem 2.1 fails in general, even when m = 0. Likewise (c) shows that 2 ⇒ 3 fails. Here HH + HH = ℓ_{ℕ}^{1}, so that ℕ \ {α} is not a set of uniqueness for HH + HH.

Let φ = (φ_{n})_{n=1}^{∞} be a multiplier of H. Then φx ∈ H, where

x = (1, 0, …, 0, −2^{n−1}, 0, …),

the −2^{n−1} being in the nth place. It follows that φ_{n} = φ_{1}. Hence all multipliers of H are constant.

Now consider ψ = (ψ_{n})_{n=1}^{∞} ∈ ℓ_{ℕ}^{∞}. It is routine to show that if f is the orthogonal projection of ψu onto u^{⊥} in ℓ^{2} then ψf^{⊥} ⊂ u^{⊥}. Hence ψ is a 1−pseudomultiplier of H. Thus the set of 1-pseudomultipliers of H contains ℓ^{∞}; it also contains many other functions. Let α ∈ ℕ and let D = ℕ \ {α}. Then D is a set of uniqueness for H, since if x ∈ H and we know x_{n} for n ≠ α we can recover x_{α} from the relation x ⊥ u. Let φ : D → ℂ be bounded and non-constant. Then φ is a 1-pseudomultiplier of H. Indeed, φk_{α}^{⊥} ⊂ H, since φx ∈ ℓ_{D}^{2} for any x ∈ H, and every element of ℓ_{D}^{2} has a unique extension to an element of H. Moreover, φ has a singularity at α in the sense of Definition 1.3. Since φ is non-constant one can construct h ∈ H such that h(α) = 0 but (φh)(α) ≠ 0. For example, if α ∉ {1, 2} and φ(1) ≠ φ(2) we may take h(1) = φ(2), h(2) = −2φ(1), h(n) = 0 for n > 2. Hence φ is a 1-pseudomultiplier of H.

We have shown that the 1-pseudomultipliers of H contain ℓ^{∞} and all non-constant bounded functions on complements of singleton subsets of ℕ. In fact these are all the pseudomultipliers of H. For let ψ be an m-pseudomultiplier. Suppose ψ is defined everywhere on ℕ except at a singularity α (the domain of ψ has to be a set of uniqueness, and so can omit at most one point). Pick a closed m-codimensional subspace E of H such that ψE ⊂ H. Then E has finite codimension as a subspace of ℓ^{2} and satisfies ψE ⊂ ℓ^{2}. It follows that ψ is bounded (see Example 1.8). It is easy to see that if ψ is constant it cannot have a singularity at α. Thus ψ is a bounded non-constant function on ℕ \ {α}. A similar argument shows that if ψ is everywhere defined on ℕ then ψ ∈ ℓ^{∞}. This completes the proof of (a).

(b) Let D = ℕ \ {α} for some α ∈ ℕ. We have H|D = ℓ_{D}^{2}, so that the multipliers of H|D are just the elements of ℓ_{D}^{∞}. However, only the constant sequences extend to multipliers of H.

(c) It will be seen below that the implication 1 ⇒ 2 of Theorem 2.1 does not depend on the domain of φ being a set of uniqueness for HH + HH, and so φ has the property described.

(d) Let ψ be a non-constant bounded function on ℕ \ {α}, where α ∈ ℕ. For any h ∈ H, ψh is an ℓ^{2} sequence indexed by ℕ\{α}, and it has a (unique) extension to an element of H. Thus ψH ⊂ H|D_{ψ}. However, ψ is not a multiplier of H since it is not defined at α. It does not even extend to a multiplier of H since it is non-constant.

Proof of Theorem 2.1 (3) ⇒ (1) is immediate from the fact that h ↦ h|D_{φ} is an isomorphism of Hilbert spaces.

(1) ⇒ (2). Let E be a closed m-codimensional subspace of H such that φE ⊂ H|D_{φ}, so that M =^{def} M_{φ,E} is a bounded linear operator from E to H.

For λ ∈ D_{φ} and h ∈ E we have

\begin{aligned}⟨M^{∗}k_{λ}, h⟩_{E} & = ⟨k_{λ}, Mh⟩_{H} = (Mh)(λ)^{−} = φ̄(λ)h̄(λ) \\ & = φ̄(λ)⟨k_{λ}, h⟩_{H} = ⟨φ̄(λ)P_{E}k_{λ}, h⟩_{E}, \end{aligned}

where P_{E} : H → E is orthogonal projection. That is,

M^{∗}k_{λ} = φ̄(λ)P_{E}k_{λ} for λ ∈ D_{φ}.

For any t ∈ ℝ and λ, μ ∈ D_{φ} we have

⟨(t^{2} − MM^{∗})k_{μ}, k_{λ}⟩ = t^{2}k(λ, μ) − ⟨M^{∗}k_{μ}, M^{∗}k_{λ}⟩
= t^{2}k(λ, μ) − ⟨φ̄(μ)P_{E}k_{μ}, φ̄(λ)P_{E}k_{λ}⟩
= t^{2}k(λ, μ) − φ(λ)φ̄(μ)⟨P_{E}k_{μ}, k_{λ}⟩
= t^{2}k(λ, μ) − φ(λ)φ̄(μ)⟨(1 − P_{E^{⊥}})k_{μ}, k_{λ}⟩
= (t^{2} − φ(λ)φ̄(μ))k(λ, μ) + φ(λ)φ̄(μ)⟨P_{E^{⊥}}k_{μ}, k_{λ}⟩.

That is,

(t^{2} − φ(λ)φ̄(μ))k(λ, μ) = ⟨(t^{2} − MM^{∗})k_{μ}, k_{λ}⟩ − φ(λ)φ̄(μ)⟨P_{E^{⊥}}k_{μ}, k_{λ}⟩.

Since P_{E^{⊥}} has rank m, the kernel −φ(λ)φ̄(μ)⟨P_{E^{⊥}}k_{μ}, k_{λ}⟩ has at most m negative squares. If t ≥ ∥M∥ the kernel ⟨(t^{2} − MM^{∗})k_{μ}, k_{λ}⟩ is positive. Thus

(t^{2} − φ(λ)φ̄(μ))k(λ, μ)

has at most m negative squares as long as t ≥ ∥M∥, in particular when t = ∥M∥ = s_{m}(φ).

(2) ⟹ (3). We begin with a simple observation that explains the introduction of the class HH + HH.

Lemma 2.3. Let M ∈ L(H), let φ : D_{φ} ⊂ Ω → ℂ be a function whose domain D_{φ} is a set of uniqueness of both H and HH + HH, and let F be an m-dimensional subspace of H, where 0 ≤ m < ∞. Suppose that, for all h ∈ F^{⊥} and λ ∈ D_{φ},

(Mh)(λ) = φ(λ)h(λ).

Then the subset S = {λ : k_{λ} ∈ F } of Ω contains at most m points, and there is a function ψ : Ω \ S → ℂ such that ψ(λ) = φ(λ) for λ ∈ D_{φ} \ S and

(Mh)(λ) = ψ(λ)h(λ) (2.2)

for all λ ∈ Ω \ S and h ∈ F^{⊥}.

Proof. Since k is nonsingular, the k_{λ} are linearly independent, and so S can contain at most m points. Consider any λ ∈ Ω \ S. Pick an h ∈ F^{⊥} such that h(λ) ≠ 0 (this is possible, since otherwise h ∈ F^{⊥} implies h ∈ k_{λ}^{⊥}, i.e., k_{λ} ∈ F). Define ψ(λ) to be (Mh)(λ)/h(λ). This definition is independent of the choice of h ∈ F^{⊥} such that h(λ) ≠ 0. For suppose h_{1}, h_{2} are two choices. Then for any μ ∈ D_{φ} we have

(h_{1} · Mh_{2})(μ) − (Mh_{1} · h_{2})(μ) = h_{1}(μ)φ(μ)h_{2}(μ) − φ(μ)h_{1}(μ)h_{2}(μ) = 0.

Since D_{φ} is a set of uniqueness for HH + HH it follows that h_{1} · Mh_{2} − Mh_{1} · h_{2} = 0, and so

\frac{Mh_{1}(λ)}{h_{1}(λ)} = \frac{Mh_{2}(λ)}{h_{2}(λ)} .

Thus ψ : Ω \ S → ℂ is well defined.

Now consider λ ∈ Ω \ S and g ∈ F^{⊥}. Choose h ∈ F^{⊥} such that h(λ) ≠ 0. The same argument as above shows that h · Mg − Mh · g = 0. Hence

(Mg)(λ) = \frac{(Mh)(λ)}{h(λ)}g(λ) = ψ(λ)g(λ).

Thus (2.2) is satisfied. Pick λ ∈ D_{φ} \ S and h ∈ F^{⊥} such that h(λ) ≠ 0. Then

φ(λ)h(λ) = (Mh)(λ) = ψ(λ)h(λ), and so φ(λ) = ψ(λ).

Example 1.2 shows that S can contain points of D_{φ}.

For the remainder of this section denote by F the set of finite subsets of D_{φ}. For F ∈ F, let

ℳ_{F} = span{k_{λ} : λ ∈ F }

and let R_{φ} be the closed linear span of {φ̄(λ)k_{λ} : λ ∈ D_{φ}}.

Define T_{F} ∈ L(ℳ_{F}) by

T_{F} k_{λ} = φ̄(λ)k_{λ}, for λ ∈ F,

and note the following standard calculation. For any x = ∑_{λ∈F} c_{λ}k_{λ} ∈ ℳ_{F} we have

∥x∥^{2} − ∥T_{F} x∥^{2} = c^{∗}((1 − φ(λ)φ̄(μ))k(λ, μ))c,

where c = [c_{λ}]_{λ∈F}, regarded as a column vector. It follows that, if φ is an mpseudomultiplier and s_{m}(φ) ≤ 1, then 1 − T_{F}^{∗}T_{F} has at most m negative eigenvalues.

Lemma 2.4. Let φ satisfy condition 2 in Theorem 2.1 and let t_{0} be the least value of t ≥ 0 such that the kernel (2.1) has at most m negative squares on D_{φ}. There exist functions

a_{j} : D_{φ} → ℂ, for 1 ≤ j ≤ m,

and a positive operator B on H such that ∥B∥ = t_{0}^{2}, BH ⊂ R_{φ}, and

⟨Bk_{λ}, k_{μ}⟩ = φ̄(λ)φ(μ)⟨k_{λ}, k_{μ}⟩ − ∑_{j=1}^{m} ā_{j}(λ)a_{j}(μ) for all λ, μ ∈ D_{φ}. (2.3)

Proof. We can assume t_{0} = 1. For any F ∈ F, the operator 1 − T_{F}^{∗}T_{F} has at most m negative eigenvalues. Hence there exists a contraction B_{F} on ℳ_{F} with 0 ≤ B_{F} ≤ 1 and vectors u_{1}^{F} , …, u_{m}^{F} ∈ ℳ_{F} such that

T_{F}^{∗}T_{F} = B_{F} + ∑_{j=1}^{m} u_{j}^{F} ⊗ u_{j}^{F} . (2.4)

The u_{j}^{F} can be taken to be suitably scaled Schmidt vectors of T_{F} (some could be zero). Note that (2.4) is equivalent to

φ̄(λ)φ(μ)⟨k_{λ}, k_{μ}⟩ = ⟨B_{F} k_{λ}, k_{μ}⟩ + ∑_{j=1}^{m} u_{j}^{F} (λ)^{−} u_{j}^{F} (μ) (2.5)

for all F ∈ F and λ, μ ∈ F. In particular,

∑_{j=1}^{m} |u_{j}^{F} (λ)|^{2} = |φ(λ)|^{2}∥k_{λ}∥^{2} − ⟨B_{F} k_{λ}, k_{λ}⟩ ≤ |φ(λ)|^{2}∥k_{λ}∥^{2} for λ ∈ F ∈ F. (2.6)

If φ(λ) = 0 for some λ, then

∑_{j=1}^{m} |u_{j}^{F} (λ)|^{2} = −⟨B_{F} k_{λ}, k_{λ}⟩,

and we must have B_{F} k_{λ} = 0. Hence B_{F} ℳ_{F} ⊂ R_{φ} for all F ∈ F.

Let B_{F}^{♯} = B_{F} ⊕ O_{M_{F}^{⊥}} ∈ L(H). Then 0 ≤ B_{F}^{♯} ≤ 1 and B_{F}^{♯} H ⊂ R_{φ} for all F ∈ F. For each F ∈ F let

Q_{F} = {G ∈ F : F ⊂ G}.

No Q_{F} is empty, and Q_{F_{1}} ∩ Q_{F_{2}} = Q_{F_{1}∪F_{2}}. Thus {Q_{F} : F ∈ F} is a filter base on F, generating a filter Q. Let U be an ultrafilter on F that refines Q. Let α be the mapping from F to the closed unit ball B of L(H) given by α(F) = B_{F}^{♯}. One can think of α as a continuous mapping from F with the discrete topology to B with the weak operator topology. B is compact and so α extends in a unique way to a continuous mapping from the Stone-Čech compactification βF of F into B. Since U can be identified with a point of βF it follows that

B = lim_{F →U} α(F) = lim_{F →U} B_{F}^{♯} (2.7)

exists in B. Clearly 0 ≤ B ≤ 1 and BH ⊂ R_{φ}. Recall that u_{j}^{F} (λ) is defined when λ ∈ D_{φ}, 1 ≤ j ≤ m and λ ∈ F ∈ F. With the same range of λ and j, extend the definition to all F ∈ F by setting u_{j}^{F} (λ) = 0 if λ ∉ F. For fixed j and λ ∈ D_{φ} the complex-valued function F ↦ u_{j}^{F} (λ) is bounded on F, by (2.6). Hence

a_{j}(λ) =^{def} lim_{F →U} u_{j}^{F} (λ)

exists for all λ ∈ D_{φ} and 1 ≤ j ≤ m. Again by (2.6) we have, for all λ ∈ D_{φ},

∑_{j=1}^{m} |a_{j}(λ)|^{2} ≤ |φ(λ)|^{2}∥k_{λ}∥^{2}. (2.8)

We now have to show that we can take limits in (2.5) to obtain (2.3).

Fix λ, μ ∈ D_{φ} and let ε > 0. By virtue of (2.8) and (2.6) there exists U ∈ U such that

|∑_{j} a_{j}(λ)^{−} a_{j}(μ) − ∑_{j} u_{j}^{F} (λ)^{−} u_{j}^{F} (μ)| < \frac{ε}{2} (2.9)

whenever F ∈ U. Similarly, there exists V ∈ U such that

|⟨Bk_{λ} − B_{F}^{♯} k_{λ}, k_{m}⟩| < \frac{ε}{2}

whenever F ∈ V. Pick F ∈ U ∩ V ∩ Q_{{λ}} (this set belongs to U, hence is nonempty). Then λ ∈ F, and so B_{F}^{♯} k_{λ} = B_{F} k_{λ}. Hence

|⟨Bk_{λ} − B_{F} k_{λ}, k_{μ}⟩| < \frac{ε}{2}.

Since F ∈ U, inequality (2.9) is also satisfied. Combining these inequalities with (2.5) we obtain

|φ̄(λ)φ(μ)⟨k_{λ}, k_{μ}⟩ − ⟨Bk_{λ}, k_{μ}⟩ − ∑_{j} ā_{j}(λ)a_{j}(μ)| < ε.

Since ε was arbitrary, (2.3) is satisfied.

It remains to show that ∥B∥ ≥ t_{0}^{2}. By (2.3), for any t ∈ ℝ and λ, μ ∈ D_{φ},

(t^{2} − φ(μ)φ̄(λ))k(μ, λ) = ⟨(t^{2} − B)k_{λ}, k_{μ}⟩ − ∑_{j=1}^{m} a_{j}(μ)ā_{j}(λ).

If t^{2} = ∥B∥ then the first term on the right hand side is positive and the second has m negative squares. Thus (t^{2} − φ(μ)φ̄(λ))k(μ, λ) has m negative squares, and so t_{0}^{2} ≤ t^{2} = ∥B∥.

In Example 1.1 (λ^{−m} on 𝔻) one can take each B_{F} to be 1_{ℳ_{F}} , hence B = 1_{H}, and one finds that a_{1}(λ) = λ^{−m}. In Example 1.2 (the characteristic function of a point) it transpires that B = 0 and a_{1} = φ. In both cases a_{j} fails to belong to H.

Lemma 2.5. For φ as in Lemma 2.4 there exist A and L in L(H) such that

(1) the range and cokernel of both A and L are contained in R_{φ};

(2) 0 ≤ A ≤ 1 and rank A ≤ m;

(3) ∥L∥ = t_{0} and, for every λ ∈ D_{φ} and h ∈ H,

(Lh)(λ) = φ(λ)(h(λ) − (Ah)(λ)) (2.10)

and

L^{∗}k_{λ} = (1 − A)φ̄(λ)k_{λ}. (2.11)

Proof. The relation (2.3) in Lemma 2.4 can be written

⟨φ̄(λ)k_{λ}, φ̄(μ)k_{μ}⟩_{H} = ⟨B^{1/2}k_{λ} ⊕ (ā_{1}(λ), …, ā_{m}(λ)), B^{1/2}k_{μ} ⊕ (ā_{1}(μ) …, ā_{m}(μ))⟩_{H⊕ℂ^{m}}

for all λ, μ ∈ D_{φ}. Hence we can define an isometry V : R_{φ} → R_{φ} ⊕ ℂ^{m} by

V φ̄(λ)k_{λ} = B^{1/2}k_{λ} ⊕ (ā_{1}(λ), …, ā_{m}(λ)) for λ ∈ D_{φ}. (2.12)

The isometry V necessarily has the form

V = [ C ; 1 ⊗ f_{1} ; ⋮ ; 1 ⊗ f_{m}] (2.13)

for some C : R_{φ} → R_{φ} and f_{1}, …, f_{m} ∈ R_{φ}. The relation V^{∗}V = 1 yields

C^{∗}C + f_{1} ⊗ f_{1} + ⋯ + f_{m} ⊗ f_{m} = 1.

Let

A = f_{1} ⊗ f_{1} + ⋯ + f_{m} ⊗ f_{m}.

Then 0 ≤ A ≤ 1, AH ⊂ R_{φ}, rank A ≤ m and C^{∗}C = 1 − A. From (2.12) and (2.13) we have

\begin{aligned}Cφ̄(λ)k_{λ} & = B^{1/2}k_{λ} \\ φ̄(λ)f̄_{j}(λ) & = ā_{j}(λ) \end{aligned} (2.14)

for λ ∈ D_{φ} and 1 ≤ j ≤ m. Extend C to an operator C^{♯} on H by taking the direct sum of C and the zero operator on R_{φ}^{⊥}, and define

L = B^{1/2}C^{♯} : H → H. (2.15)

Clearly the range and cokernel of L are contained in R_{φ}. Since ∥B^{1/2}∥ = t_{0} and ∥C∥ ≤ 1 we have

∥L∥ ≤ t_{0}.

On applying C^{∗} to (2.14) we find that, for λ ∈ D_{φ},

L^{∗}k_{λ} = (C^{♯})^{∗}B^{1/2}k_{λ} = C^{∗}B^{1/2}k_{λ} = C^{∗}Cφ̄(λ)k_{λ} = (1 − A)φ̄(λ)k_{λ}.

Hence, for any h ∈ H and λ ∈ D_{φ},

\begin{aligned}(Lh)(λ) & = ⟨Lh, k_{λ}⟩ = ⟨h, L^{∗}k_{λ}⟩ = ⟨h, (1 − A)φ̄(λ)k_{λ}⟩ \\ & = φ(λ)⟨(1 − A)h, k_{λ}⟩ = φ(λ)(h(λ) − (Ah)(λ)). \end{aligned}

It remains to show that ∥L∥ ≥ t_{0}, or equivalently, that the kernel

(∥L∥^{2} − φ̄(λ)φ(μ)) k(μ, λ)

has at most m negative squares. From (2.11) we have, for λ, μ ∈ 𝔻_{φ},

\begin{aligned}⟨L^{∗}k_{λ}, L^{∗}k_{μ}⟩ & = ⟨(1 − A)φ̄(λ)k_{λ}, (1 − A)φ̄(μ)k_{μ}⟩ \\ & = φ̄(λ)φ(μ)⟨(1 − 2A + A^{2})k_{λ}, k_{μ}⟩. \end{aligned}

Hence

(∥L∥^{2} − φ̄(λ)φ(μ)) k(μ, λ) = ⟨(∥L∥^{2} − LL^{∗})k_{λ}, k_{μ}⟩+
φ̄(λ)φ(μ) [⟨(1 − 2A + A^{2})k_{λ}, k_{μ}⟩ − ⟨k_{λ}, k_{μ}⟩]
= a positive kernel − φ̄(λ)φ(μ)⟨A(2 − A)k_{λ}, k_{μ}⟩.

Since A(2 − A) is positive and has rank ≤ m, the right hand side has at most m negative squares, as required.

We can now prove that (2) ⇒ (3) in Theorem 2.1. Suppose that φ satisfies (2), the least value of t being t_{0}. By Lemma 2.5 there exist A and L in L(H) such that ||L|| = t_{0} and, for all λ ∈ D_{φ} and h ∈ H,

Lh(λ) = φ(λ)(h(λ) − (Ah)(λ)).

Moreover A has rank m, so that Ker A has codimension m, and for all h ∈ Ker A and λ ∈ D_{φ},

Lh(λ) = φ(λ)h(λ).

By Lemma 2.3 there is a subset S of Ω containing at most m points and a function ψ : Ω \ S → ℂ such that ψ(λ) = φ(λ) for λ ∈ D_{φ} \ S and

ψ(λ)h(λ) = Lh(λ) = φ(λ)h(λ) (2.16) for all h ∈ Ker A and λ ∈ Ω \ S. Extend ψ to a function on D_{ψ} = (Ω \ S) ∪ D_{φ} by defining ψ(λ) = φ(λ) for λ ∈ D_{φ} ∩ S. Then ψ extends φ and its domain D_{ψ} is a set of uniqueness for H (since D_{ψ} ⊃ D_{φ}). There are at most m points in Ω \ D_{ψ} ⊂ S, and (2.16) tells us that ψKer A ⊂ H. Thus ψ is an m-pseudomultiplier of H which extends φ. Hence (2) ⇒ (3).

Now suppose (1)-(3) hold. From (2.16) we have

s_{m}(ψ) ≤ ||L|| = t_{0}.

In the proof of (1) ⇒ (2) we showed that the kernel (2.1) has at most m negative squares if t = s_{m}(φ), that is,

t_{0} ≤ s_{m}(φ).

If E is an m-codimensional subspace of H such that ψE ⊂ H then φE|D_{φ} ⊂ H|D_{φ} and ||M_{ψ,E}|| = ||M_{φ,E}|D_{φ}||. Taking infima over closed m-codimensional subspaces E such that ψE ⊂ H we obtain

s_{m}(φ) ≤ s_{m}(ψ).

Combining the last three inequalities, we have

s_{m}(φ) = t_{0} = s_{m}(ψ).

We can now derive a simple but informative multiplicity result for pseudomultipliers.

Theorem 2.6. Let H be a Hilbert space of functions with nonsingular kernel on a set Ω and let φ be an m-pseudomultiplier of H. For any ξ ∈ ℂ such that |ξ| > s_{m}(φ), the equation φ(λ) = ξ has at most m solutions for λ ∈ D_{φ}.

Proof. Pick L, A as in Lemma 2.5. Suppose there are n points λ_{1}, …, λ_{n} ∈ D_{φ} which are solutions of the equation, where n > m. Since A has rank at most m there exist scalars c_{1}, …, c_{n}, not all zero, such that ∑ c_{j}Ak_{λ_{j}} = 0. Let h = ∑ c_{j}k_{λ_{j}}. Then h ≠ 0, and

L^{∗}h = ∑ c_{j}L^{∗}k_{λ_{j}} = ∑ c_{j}ξ̄(k_{λ_{j}} − Ak_{λ_{j}}) = ξ̄h,

and so ξ̄ is an eigenvalue of L^{∗}. However

||L^{∗}|| = s_{m}(φ) < |ξ|,

a contradiction.

We conclude this section with an aside about the key Lemma 2.5. How should we understand the rank m Hermitian operator A associated with an m-pseudomultiplier φ? Let us consider the case m = 1, and put A = f ⊗ f for some f ∈ H. The relation (2.10) becomes

(Lh)(λ) = φ(λ) (h(λ) − ⟨h, f⟩f(λ)) (2.17)

for h ∈ H, λ ∈ D_{φ}. What is the connection between the pseudomultiplier φ and the function f? We propose that f be regarded as a singular vector of the (unbounded) operator M_{φ} on H corresponding to the singular value s_{0}(φ) = ∞. Strictly speaking this does not make sense, but we can give it a meaning as follows.

Consider first the case of a finite kernel k on Ω = {λ_{1}, …, λ_{n}}, and define T on H by T k_{λ} = φ̄(λ)k_{λ} (so that T = T_{Ω} in the notation above). Then T has a maximising vector u such that, for some contraction B on H,

T^{∗}T = B + u ⊗ u.

Here ||u|| = ||T ||, and in the notation of Lemma 2.4, u = a_{1}. By equation (2.14), φf = u, and so

f = \frac{1}{φ}u = (T^{∗})^{−1}u.

Thus u = T^{∗}f. Hence f is a maximising vector of T^{∗} and ||f|| = 1. In other words f is a singular vector of the multiplication operator M_{φ} corresponding to s_{0}(φ). For finite kernels, then, our statement about f is meaningful and true.

Now consider a nonsingular kernel on a general set Ω and apply the foregoing observation to T_{F} for each F ∈ F. Then there is a unit maximising vector f_{F} ∈ M_{F} and a contraction B_{F} on M_{F} such that

T_{F}^{∗}T_{F} = B_{F} + T_{F}^{∗}f_{F} ⊗ T_{F}^{∗}f_{F} .

Along the ultrafilter U, f_{F} tends weakly to a limit, which must equal f. If ||f|| < 1 then φ is a multiplier of H (put h = f in (2.17)). If ||f|| = 1 then we have f_{F} → f in norm as F → U. That is, f is the norm limit of unit maximising vectors of the approximating finite rank compressions T_{F}^{∗}, F ∈ U of the multiplication operator M_{φ}.

3. Pseudomultipliers of analytic kernels

What are the pseudomultipliers of the familiar Hilbert spaces of analytic functions? The examples we began with (1/z and the characteristic function of a point) might lead one to the optimistic hope that the 1-pseudomultipliers are simply the multipliers modified by a removable singularity or a simple pole. We already know that, for a general analytic kernel, this can fail to be true in at least three ways: the domain Ω could be too small (Example 1.4), H could really be a space of functions of z^{2} (Example 1.10) or there could be more subtle structural reasons (Example 1.11). Nevertheless, it is true for a wide class of kernels, as we show in Theorem 3.3 below.

Let Ω be a domain in ℂ and let H be a Hilbert space of analytic functions on Ω with reproducing kernel k. Such a kernel will be called an analytic kernel. There is one simple observation we can make about pseudomultipliers of such kernels.

Theorem 3.1. Let k be a nonsingular analytic kernel on a domain Ω and let φ be an m-pseudomultiplier of the associated Hilbert space H of functions on Ω. Let the defect of φ be d. There is a set S ⊂ D_{φ}, containing at most m − d points, such that φ is analytic at every point of D_{φ} \ S. Moreover, each point of Ω \ D_{φ} is a pole of φ and each point of S is either a removable singularity or a pole of φ.

Note that it can happen that a point of D_{φ} is a pole of φ: recall Example 1.7.

Proof. By Lemma 2.5 there exist bounded linear operators A, L on H such that 0 ≤ A ≤ 1, A has rank at most m and, for every h ∈ H and λ ∈ D_{φ},

(Lh)(λ) = φ(λ) (h(λ) − (Ah)(λ)) .

We claim that φ is analytic at any β ∈ Ω such that k_{β} ∉ AH. Indeed, for such β, Ker A is not a subset of k_{β}^{⊥}, and so there exists h ∈ H such that Ah = 0 and h(β) ≠ 0. We have Lh = φh, and so φ = (Lh)/h on any neighbourhood of β on which h does not vanish. Thus φ is analytic at β.

It follows that k_{β} ∈ AH for the d points of Ω \ D_{φ}. Since k is nonsingular and AH has dimension ≤ m, the set

S =^{def} {β ∈ D_{φ} : k_{β} ∈ AH}

can contain at most m − d points. The preceding paragraph shows that φ is analytic at each point of D_{φ} \ S.

Now consider any point α ∈ Ω \ D_{φ}. Then α is a singularity of φ. That is, there exist u, g ∈ H such that u(α) = 0, φu = g|D_{φ} and g(α) ≠ 0. Clearly u ≠ 0, so that α is an isolated zero of u. Hence we can write φ = g/u on a punctured neighbourhood of α, and so φ has a pole at α.

Consider any point α ∈ S. Since φ is analytic in a punctured neighbourhood of α it must be the case that α is either a removable singularity, a pole or an essential singularity of φ. In the latter case, by the great Picard theorem, φ attains all but one complex values infinitely often, contradicting Theorem 2.6. Hence one of the other two alternatives holds.

This is an opportune moment to point out that pseudomultipliers of Hilbert spaces of analytic functions of several complex variables are not interesting objects, at least as we have defined them here. By exactly the argument above, an m-pseudomultiplier φ of such a space is analytic at all but at most m points. Now a singularity of φ in the sense of Definition 1.3 would have to be an isolated pole of φ, and an analytic function of several variables cannot have any such. Thus the m points where analyticity fails are removable singularities, and φ is just an analytic function plus some point discontinuities. However, it is possible the notion of pseudomultiplier could be developed for Hilbert spaces of vector-valued functions so as to apply to analytic functions of several variables.

Corollary 3.2. Let H be a Hilbert space of entire functions having nonsingular kernel on ℂ, and let φ be an m-pseudomultiplier of defect d on H. There exists a rational function ψ of degree at most m which agrees with φ at all except at most m−d points of D_{φ}.

Here the degree of ψ is defined to be the sum of the multiplicities of the poles of ψ, including ∞ if applicable. The points at which φ and ψ differ can include poles of ψ.

Proof. By Theorem 3.1 there is a set S ⊂ D_{φ} containing at most m − d points, consisting of poles and removable singularities of φ, such that φ is analytic on D_{φ} \S. Let ψ be the function obtained from φ by removing any removable singularities and deleting from the domain of φ any point α ∈ S which is a pole of φ|D_{φ} \ {α}. Then ψ is meromorphic in the whole complex plane. Again by the great Picard theorem and Theorem 2.6, ψ cannot have an essential singularity at ∞ and hence is rational. Clearly ψ agrees with φ except at points of S.

In order to get more detailed information about the nature of pseudomultipliers of analytic kernels we consider kernels satisfying the following two axioms: (AK1): H is invariant under the operation M_{z} of multiplication by the independent

variable; (AK2): Every bounded linear operator on H which commutes with M_{z} is a multi-

plication operator. Not all spaces of interest have these properties (e.g., the Fock space does not satisfy

(AK1)), but many of them do. An easy consequence of (AK1) is

M_{z}^{∗}k_{λ} = λ̄k_{λ} (3.1)

for all λ ∈ Ω.

Remark. (AK2) holds if (AK1) is true and either H contains the polynomials on Ω as a dense subspace or every eigenvector of M_{z}^{∗} has the form ck_{α} for some c ∈ ℂ and α ∈ Ω.

For suppose T M_{z} = M_{z}T. In the former case it is plain that T is multiplication by M1. In the latter we have, for any λ ∈ Ω,

M_{z}^{∗}T^{∗}k_{λ} = T^{∗}M_{z}^{∗}k_{λ} = λ̄T^{∗}k_{λ}

and so T^{∗}k_{λ} is either 0 or an eigenvector of M_{z}^{∗} with eigenvalue λ̄. Hence, by assumption,

T^{∗}k_{λ} = ψ̄(λ)k_{λ}

for some ψ(λ) ∈ ℂ. For any h ∈ H and λ ∈ Ω

(T h)(λ) = ⟨T h, k_{λ}⟩ = ⟨h, T^{∗}k_{λ}⟩ = ⟨h, ψ̄(λ)k_{λ}⟩ = ψ(λ)h(λ).

Thus ψ is a multiplier and T = M_{ψ}.

Theorem 3.3. Let H be a Hilbert space of analytic functions on a domain Ω. Suppose that H has nonsingular kernel and that (AK1) and (AK2) hold. Let φ : D_{φ} ⊂ Ω → ℂ be a 1-pseudomultiplier of H. Then there exists α ∈ ℂ and a multiplier θ of H such that D_{φ} is either Ω or Ω \ {α} and

φ(λ) = \frac{θ(λ)}{λ − α} for λ ∈ Ω \ \{α\}.

We do not assert that α ∈ Ω in general. Observe that the conclusion holds when φ is a multiplier or even a function that differs from a multiplier at a single point α ∈ Ω (take θ(λ) = (λ − α)φ(λ)).

Proof. By Lemma 2.5 there exist L ∈ L(H) such that ∥L∥ = s_{1}(φ) and f ∈ H such that ∥f∥ ≤ 1 and, for all λ ∈ D_{φ},

L^{∗}k_{λ} = φ̄(λ)k_{λ} − φ̄(λ)f̄(λ)f

(we have put A = f ⊗ f). From this equation and (3.1) it follows that

(M_{z}^{∗}L^{∗} − L^{∗}M_{z}^{∗})k_{λ} = φ̄(λ)f̄(λ)(λ̄f − M_{z}^{∗}f) (3.2)

for all λ ∈ D_{φ}. Hence

Range(M_{z}^{∗}L^{∗} − L^{∗}M_{z}^{∗}) ⊂ span{f, M_{z}^{∗}f}.

We consider the three possible dimensions of this span.

Suppose f = 0: then by 2.10, for λ ∈ D_{φ} and h ∈ H,

(Lh)(λ) = φ(λ)h(λ).

By Lemma 2.3 and Theorem 1.13 D_{φ} = Ω and φ is a multiplier of H. (Incidentally, we have in this case

s_{0}(φ) = ||M_{φ}|| = ||L|| = s_{1}(φ),

so that s_{0}(φ) is a multiple singular value.)

Secondly, suppose f and M_{z}^{∗}f span a space of dimension one: say M_{z}^{∗}f = ᾱf for some α ∈ ℂ. In equation (2.10) replace h by (M_{z} − α)h to obtain, for all λ ∈ D_{φ} and h ∈ H,

(L(M_{z} − α)h)(λ) = φ(λ) ((λ − α)h(λ) − ⟨(M_{z} − α)h, f⟩f(λ))
= φ(λ) ((λ − α)h(λ) − ⟨h, M_{z}^{∗}f − ᾱf⟩f(λ))
= (λ − α)φ(λ)h(λ).

Since L(M_{z} − α) is bounded, Lemma 2.3 tells us that there is a multiplier θ of H such that

θ(λ) = (λ − α)φ(λ) for all λ ∈ D_{φ}. (3.3)

We claim that φ can have no singularity in Ω other than α. For suppose β ≠ α is a singularity. Then there exists h ∈ H such that h(β) = 0 and φh has an extension u ∈ H such that u(β) ≠ 0. Now (M_{z} − α)u ∈ H is an extension of the function λ ↦ (λ − α)(φh)(λ) on D_{φ}, that is, of θh|D_{φ}. Since θh ∈ H, it follows that θh = (M_{z} − α)u. Evaluating both sides at β we have

0 = θ(β)h(β) = (β − α)u(β) ≠ 0,

a contradiction. Thus β cannot be a singularity, and so D_{φ} is either Ω or Ω \ {α}. It follows that D_{φ} \ {α} = Ω \ {α}, and, from (3.3),

φ(λ) = \frac{θ(λ)}{λ − α} for λ ∈ Ω \ \{α\}

as required.

The remaining possibility is that span{f, M_{z}^{∗}f} has dimension 2. Then, by (3.2), there exist u, v ∈ H such that

M_{z}^{∗}L^{∗} − L^{∗}M_{z}^{∗} = f ⊗ u − M_{z}^{∗}f ⊗ v. (3.4)

On applying both sides to k_{λ} we find, for λ ∈ D_{φ},

φ̄(λ)f̄(λ)(λ̄f − M_{z}^{∗}f) = ū(λ)f − v̄(λ)M_{z}^{∗}f.

By the linear independence of f and M_{z}^{∗}f,

λ̄φ̄(λ)f̄(λ) = ū(λ), φ̄(λ)f̄(λ) = v̄(λ)

on D_{φ}. It follows that u − M_{z}v vanishes on D_{φ}, hence is zero. Thus (3.4) becomes

M_{z}^{∗}L^{∗} − L^{∗}M_{z}^{∗} = (f ⊗ v)M_{z}^{∗} − M_{z}^{∗}(f ⊗ v).

Consequently L + v ⊗ f commutes with M_{z}. Thus there exists a multiplier θ of H such that L + v ⊗ f = M_{θ}. Then, for λ ∈ D_{φ},

θ̄(λ)k_{λ} = M_{θ}^{∗}k_{λ} = L^{∗}k_{λ} + (f ⊗ v)k_{λ} = φ̄(λ)k_{λ} − φ̄(λ)f̄(λ)f + v̄(λ)f = φ̄(λ)k_{λ}.

Thus φ is the restriction of a multiplier. By Theorem 1.13, φ is a multiplier.

It would be of interest to identify the pseudomultipliers and the corresponding quantities s_{m}(·) for favourite function spaces. As we have mentioned, in the case of H^{2}, a well known theorem of Adamyan, Arov and Krein provides a complete description. To present it we introduce some terminology. Suppose that φ, ψ are functions with domains D_{φ}, D_{ψ} contained in a set Ω. We shall say that ψ is a finite modification of φ if D_{ψ} ⊃ D_{φ}, D_{ψ} \ D_{φ} is finite and φ, ψ agree at all but finitely many points of D_{φ}. We recall that H_{(ℓ)}^{∞} denotes the set of functions of the form φ = f/p where f ∈ H^{∞} and p is a polynomial with at most ℓ zeros in 𝔻, counting multiplicities, and none of unit modulus; we take D_{φ} to be the complement of the set of poles of φ in 𝔻 (thus we suppose that all removable singularities have been removed).

Theorem 3.4. A function φ : D_{φ} ⊂ Ω → ℂ is a pseudomultiplier of H^{2} if and only if there is a function ψ ∈ H_{(ℓ)}^{∞}, for some non-negative integer ℓ, such that φ is a finite modification of ψ. Moreover, if φ is an m-pseudomultiplier of H^{2}, then s_{m}(φ) = ∥φ∥_{L^{∞}(𝕋)}.

This is essentially the main result of [AAK], where however it is expressed in a rather different context. A careful derivation of this form of the theorem from the original one can be found in [Q2, Sec. 7].

We shall conclude with a description of the pseudomultipliers of the Fock space (see Example 1.9). Recall that a rational function is said to be proper if it has a finite limit at infinity.

Theorem 3.5. The pseudomultipliers of the Fock space Φ are precisely the finite modifications of the proper rational functions.

Proof. Let E be the space of all entire functions. We claim that if p is a non-zero polynomial, f is an entire function and pf ∈ Φ then f ∈ Φ. Indeed, if p is nonconstant, {z : |p(z)| < 1} is a bounded subset of ℂ, and so

\begin{aligned}∫_{ℂ} |f(z)|^{2}e^{−|z|^{2}}dA & = ∫_{|p(z)|<1} + ∫_{|p(z)|>1} |f(z)|^{2}e^{−|z|^{2}}dA \\ & ≤ ∫_{|p(z)|<1} |f(z)|^{2}e^{−|z|^{2}}dA + ∫_{|p(z)|>1} |p(z)f(z)|^{2}e^{−|z|^{2}}dA < ∞. \end{aligned}

Let φ be a proper rational function of degree n with denominator p. We show that φ is a pseudomultiplier of Φ. Let E = Φ ∩ pE. Then E has codimension n in Φ: it is given by the vanishing of n linear functionals, which are evaluations of f and some of its derivatives at zeros of p. Write φ as a sum of partial fractions in which each term is either constant or is a constant divided by a divisor of p. The claim above shows that each such term multiplies E into Φ and hence φE ⊂ Φ. It is clear that each pole of φ is a singularity in the sense of Definition 1.3, and hence φ is an n-pseudomultiplier, whose domain D_{φ} is the complement of the set of poles of φ. Now consider the function ψ obtained by changing the value of φ at points α_{1}, …, α_{r} ∈ D_{φ} and giving it a value at poles β_{1}, …, β_{s} of φ. Then ψ has domain D_{ψ} = D_{φ} ∪ {β_{1}, …, β_{s}}. If β_{j} has multiplicity n_{j} then ψ multiplies the closed (n + r + s)-codimensional subspace

{f ∈ Φ ∩ pE : f(α_{j}) = 0, 1 ≤ j ≤ r, f^{(n_{i})}(β_{i}) = 0, 1 ≤ i ≤ s}

of Φ into Φ|D_{ψ}. Points outside the domain of ψ are poles of φ and they remain singularities of ψ. Thus ψ is an (n + r + s)-pseudomultiplier of Φ. That is, any finite modification of a proper rational function is a pseudomultiplier of Φ.

Conversely, consider an m-pseudomultiplier ψ of Φ. Let ψ have defect d. By Corollary 3.2 there exists a rational function φ which differs from ψ at no more than m − d points. We have to show that φ is proper.

The function z is not a pseudomultiplier of Φ. For suppose s_{m}(z) = t < ∞. Then by Theorem 2.1 the kernel (t^{2} − λμ̄)e^{λμ̄} has at most m negative squares. However,

(t^{2} − w)e^{w} = t^{2} + ∑_{n=1}^{∞} (\frac{t^{2}}{n} − 1) \frac{w^{n}}{(n − 1)!},

which has infinitely many negative coefficients, a contradiction. It follows that z ↦ z − a is not a pseudomultiplier, for any a ∈ ℂ. In fact no non-constant polynomial is a pseudomultiplier of Φ. For suppose that p is such a polynomial and pE ⊂ Φ for some closed finite-codimensional space E ⊂ Φ. Factorize p as p(z) = (z − a)q(z) for some a ∈ ℂ and polynomial q. By the claim above, if (z − a)qE ⊂ Φ then (z − a)E ⊂ Φ, contradicting the fact that z − a is not a pseudomultiplier. Hence p is not a pseudomultiplier.

Return to the rational pseudomultiplier φ of Φ and expand it in partial fractions:

φ = p + ∑ u_{j}

where p is a polynomial and each u_{j} is a constant divided by a power of a linear function. We showed above that each u_{j} is a pseudomultiplier, and so φ − ∑ u_{j} is a pseudomultiplier. That is, p is a pseudomultiplier. It follows that p is constant, and hence that φ is proper.

References

[AAK] V. M. Adamyan, V. Z. Arov and M. G. Krein, Analytic properties of Schmidt pairs of a Hankel operator and generalized Schur–Takagi problem, Mat. Sbornik 86 (1971), 33–73.

[Ag1] J. Agler, Interpolation, preprint (1987).

[Ag2] J. Agler, Nevanlinna–Pick interpolation on Sobolev space, Proc. Amer. Math. Soc. 108(1990) 341-351.

[Ar] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.

[BB] F. Beatrous and J. Burbea, Positive definiteness and its applications to interpolation problems for holomorphic functions, Trans. Amer. Math. Soc.,284 (1984) 247–270.

[H] H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993) 1–9.

[HRS] H. Hedenmalm, S. Richter and K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman space, preprint 1995.

[P] G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1961), 7–23.

[Q1] P. Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integral Equations and Operator Theory 16 (1993), 244–266.

[Q2] P. Quiggin, Generalisations of Pick’s Theorem to Reproducing Kernel Hilbert Spaces, Ph. D. thesis, Lancaster University, 1994.

[S] F. Szafraniec, On bounded holomorphic interpolation in several variables, Monatshefte Math. 101 (1986) 59-66.

J. Agler, Department of Mathematics, University of California, San Diego, La Jolla CA 92093

N. J. Young, Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England

  E-mail address: n.young@lancaster.ac.uk