REDUCING ESSENTIAL EIGENVALUES IN THE BOUNDARY OF THE NUMERICAL RANGE

NORBERTO SALINAS AND MARIA VICTORIA VELASCO

Abstract. We examine a purely geometric property of a point in the boundary of the numerical range of a (Hilbert space) operator that implies that such a point is a reducing essential eigenvalue of the given operator. Roughly speaking, such a property means that the boundary curve of the numerical range has infinite curvature at that point (we must exclude however linear verteces because they may be reducing eigenvalues without being reducing essential eigenvalues). This result allows us to give an elegant proof of a conjecture of Joel Anderson: A compact perturbation of a scalar multiple of the identity operator can not have the closure of its numerical range equal to half a disk (neither equal to any acute circular sector).

In this note, we examine a purely geometric property of a point in the boundary of the numerical range of a (Hilbert space) operator which implies that such a point is a reducing essential eigenvalue of the given operator (see Theorem 1.7 and Corollary 1.8). Roughly speaking, such a property means that the boundary curve of the numerical range has infinite curvature at that point (we must exclude however linear verteces because they may be reducing eigenvalues without being reducing essential eigenvalues). This result allows us to give an elegant proof of a conjecture of Joel Anderson (see [1], [7], [8]):

A compact perturbation of a scalar multiple of the identity operator can not have the closure of its numerical range equal to half a disk (neither equal to any acute circular sector).

Indeed, there are two non-smooth points in the boundary of such a convex set that have infinite curvature and that are not linear verteces. So they are reducing essential eigenvalues (from Corollary 8) which is not possible for a multiple of the identity operator.

The above mentioned theorem also proves in the affirmative a strengthening of a conjecture of Mathias Hübner [6]: such points (of infinite curvature) are in the essential spectrum of the given operator.

1. Reducing Essential Eigenvalues and the Numerical Range

In this section, we recall some basic facts about the numerical range of an operator on Hilbert space, and the notion of reducing essential eigenvalues.

Research at MSRI is supported in part by MSRI grant DMS-9022140.

Definition 1.1. Let T be an operator acting on a (fixed) Hilbert space H. The numerical range and the state numerical range of T are defined, respectively, as follows

W (T) := {(⟨T x, x⟩ : x ∈ H, ∥x∥ = 1},
W_{s}(T) := {ϕT : ϕ ∈ S(C^{∗}(T))},

where C^{∗}(T) denotes the C^{∗}-algebra generated by T and the identity operator I on H. Also, we used the notation S(A) for the state-space of the C^{∗}-algebra A.

Remark 1.2. (a) We recall [10] that W (T) is always convex, and W_{s}(T) = \overline{W (T)}.

(b) The problem of determining the class of bounded convex sets of the form W (T) for some operator T on H is still open (an easy cardinality argument shows that there are many bounded convex sets which are not in that class). In [1] it is shown that W (T) is always a borel subset of ℂ, and in [8] it is proved that the components of W (T) in the boundary of W (T) must be either singletons or conic arcs. On the other hand, it is not even known whether W (T) can be the union of the open unit disk and a non-trivial open arc in its boundary.

Definition 1.3. Let λ be a point in the boundary of a convex subset C of the complex plane (with non-empty interior). We say that λ is of infinite curvature if, after a suitable rotation and translation which identifies λ with zero, we can choose the real axis as supporting line for C (in which case, we shall say that C is in standard position) and then

lim_{α∈C^{Im α→0}^{Re α→0}} \frac{| Im α|}{Re^{2} α} = ∞.

For convenience sake, we adopt the convention that C is in standard position if it is contained in the upper-half plane, and the point of the boundary under study is the origin. If the real axis is the only tangent line to the boundary of C, then we call the origin a point of infinite lefthand curvature whenever

lim_{α∈C^{Im α→0}^{Re α→0−}} \frac{| Im α|}{Re^{2} α} = ∞,

and we call the origin a point of infinite righthand curvature whenever

lim_{α∈C^{Im α→0}^{Re α→0+}} \frac{| Im α|}{Re^{2} α} = ∞.

Remark 1.4. (a) Obviously, in order to show that a point is of infinite curvature, the suitable rotation will not be unique in general. Of course, this will not be an obstacle for deducing later some analitic properties that such a points may satisfy.

Moreover, it should be noticed that the property of having infinite curvature depends only on the graph of the function defined by the boundary of C near λ = 0. Indeed, let C be in a standard position. Then it is easily shown that λ is of infinite curvature precisely when

lim_{α∈∂C^{α→0}} \frac{Im α}{Re^{2} α} = ∞.

(b) Of course, a linear vertex in the boundary of a convex set C (i.e., a point λ in ∂C such that C can be written as the convex hull of λ and another convex set C′ ≠ C) is a point of infinite curvature of C. In fact, it readily follows that even a corner in the boundary of a convex set is a point of infinite curvature. (We recall [7] that a point λ in the boundary of a convex set C is called a corner if C is a subset of a convex set C′ where λ is a linear vertex). But, it is easy to come up with examples of points of infinite curvature where the boundary is continuously differentiable.

(c) To have infinite lefthand curvature means that C, and more precisely the graph given by ∂C, has infinite curvature on the left of zero, while to have infinite righthand curvature means that the curvature on the right of zero is infinite. It is also clear that a point may be of infinite onesided curvature but not of infinite curvature. For instance, this is the case for the origin in the convex set C := A ∪ B where A := {α ∈ ℂ : −1 ≤ Re α ≤ 0, 0 ≤ Im α ≤ 1} and B := {α ∈ ℂ : (Re α)^{3/2} ≤ Im α ≤ 1, 0 ≤ Re α ≤ 1}.

Definition 1.5. A point λ is called an reducing approximate eigenvalue of an operator T if there exists a sequence u_{n} of unit vectors in H such that

lim_{n→∞} ∥(T − λI)u_{n}∥ + ∥(T^{∗} − \overline{λ}I)u_{n}∥ = 0.

If in addition u_{n} tends weakly to zero, then λ is called a reducing essential eigenvalue. The set of all reducing approximate eigenvalues of the operator T will be denoted by R(T) and the set of reducing essential eigenvalues by R_{e}(T).

A point λ ∈ ℂ is called a normal essential eigenvalue of T if λ is in R_{e}(T) and every time that an orthonormal sequence u_{n} satisfies ∥(T −λI)u_{n}∥ → 0, we also have ∥(T^{∗} − \overline{λ}I)u_{n}∥ → 0.

Remark 1.6. (a) The properties and characterizations of R(T) and R_{e}(T) were already discussed in [9]. In particular, λ belongs to R(T) if, and only if, there exist a ∗-homomorphism ϕ : C^{∗}(T) → ℂ such that ϕT = λ and, analogously, λ is in R_{e}(T) if and only if there exist a ∗-homomorphism ϕ : C^{∗}(T) → ℂ such that ϕ(C^{∗}(T)∩K) = 0 and ϕT = λ. (Here, we are employing the usual notation K for the ideal of compat operators) Thus, R(T) (resp. R_{e}(T)) is contained in the intersection of σ(T) (resp. σ_{e}(T)) and the complex conjugate set of σ(T^{∗}) (resp. σ_{e}(T^{∗})).

(b) Let R_{00}(T) be the set of finite multiplicity reducing eigenvalues of T which are isolated points of R(T). Here, by a finite-dimensional reducing eigenvalue of T we mean a complex number λ such that null(T − λI) ∩ null(T^{∗} − \overline{λ}I) is a non trivial finite dimensional subspace. Then, it is shown in [9], Theorem 6.1, that R(T) = R_{e}(T) ∪ R_{00}(T), where the union (of course) is disjoint.

Theorem 1.7. (a) Let 0 ∈ ∂W_{s}(T) be a point of infinite curvature and consider W_{s}(T) in standard position. If u_{n} is a sequence of unit vectors in H such that ⟨T u_{n}, u_{n}⟩ → 0 then ∥T u_{n}∥ → 0 and ∥T^{∗}u_{n}∥ → 0.

(b) Let 0 ∈ ∂W_{s}(T) be a point of infinite righthand (resp. lefthand) curvature and consider W_{s}(T) in standard position. If u_{n} is a sequence of unit vectors in H such that ⟨T u_{n}, u_{n}⟩ → 0 where, for all n, ⟨T u_{n}, u_{n}⟩ is not zero and is contained in a segment of end points given by zero and some α_{0} in W_{s}(T) with Re α_{0} > 0 (resp. Re α_{0} < 0), then ∥T u_{n}∥ → 0 and ∥T^{∗}u_{n}∥ → 0.

Proof. First of all we will stablish some inequalities which will be useful to our purposes. We begin by observing that from the condition W_{s}(T) ⊂ {λ ∈ ℂ : Im λ ≥ 0} we obtain the following elementary but important inequality:

Im⟨T u, u⟩ ≥ 0, for all u in H.

Also, given arbitrary vectors u, z in H and an arbitrary α ∈ ℂ, we have:

⟨T (u + αz), u + αz⟩ = ⟨T u, u⟩ + \overline{α}⟨T u, z⟩ + α⟨T z, u⟩ + |α|^{2}⟨T z, z⟩.

Since Im⟨T (u ± αz), u ± αz⟩ ≥ 0, we see that

| Im[\overline{α}⟨T u, z⟩ + α⟨T z, u⟩]| ≤

Im[⟨T u, u⟩ + |α|^{2}⟨T z, z⟩] ≤ |⟨T u, u⟩| + |α|^{2}∥z∥^{2}∥T ∥. (1.1)

Consequently,

| Im⟨T (u + αz), u + αz⟩| ≤ 2 Im[⟨T u, u⟩ + |α|^{2}⟨T z, z⟩] ≤

2(|⟨T u, u⟩| + |α|^{2}∥T ∥∥z∥^{2}). (1.2)

On the other hand,

| Re⟨T (u + αz), u + αz⟩| ≥
| Re[\overline{α}⟨T u, z⟩ + α⟨T z, u⟩]| − | Re⟨T u, u⟩| − |α|^{2}| Re⟨T z, z⟩| ≥

| Re[\overline{α}⟨T u, z⟩ + α⟨T z, u⟩]| − |⟨T u, u⟩| − |α|^{2}|∥T ∥∥z∥^{2}. (1.3)

Now we treat the case where zero is a point of infinite curvature in ∂W_{s}(T), i.e., assertion (a). Let u_{n} be a sequence of unit vectors in H such that ⟨T u_{n}, u_{n}⟩ → 0. We may write (uniquely) T u_{n} = δ_{n}u_{n} + β_{n}v_{n} and T^{∗}u_{n} = \overline{δ_{n}}u_{n} + \overline{γ}_{n}w_{n}, where v_{n} and w_{n} are unit vectors orthogonal to u_{n}, and β_{n}, γ_{n}, δ_{n}, are given respectively by δ_{n} := ⟨T u_{n}, u_{n}⟩, β_{n} := ⟨T u_{n}, v_{n}⟩ and γ_{n} := ⟨T w_{n}, u_{n}⟩, for all n = 1, 2, . . .. Also, by multiplying by an appropriate complex number of modulus one, if needed, we may assume that Re⟨v_{n}, w_{n}⟩ ≥ 0 for all n. Let τ_{n} be a complex number with |τ_{n}|=1 and such that |\overline{τ_{n}}β_{n}+τ_{n}γ_{n}| = |β_{n}|+|γ_{n}|, n = 1, 2, . . .. Let r_{n} := |δ_{n}|^{\frac{1}{2}} if δ_{n} ≠ 0 and r_{n} := \frac{1}{n} otherwise. We define α_{n} := r_{n}τ_{n}, for all n. Since |\overline{α_{n}}β_{n} + α_{n}γ_{n}| = r_{n}(|β_{n}| + |γ_{n}|), there exist suitable complex numbers η_{n} with |η_{n}| = 1 such that \overline{α_{n}}β_{n} + α_{n}γ_{n} = r_{n}(|β_{n}| + |γ_{n}|)η_{n}, for all n. Let z_{n} := v_{n} + w_{n}. Since ⟨T u_{n}, z_{n}⟩ = β_{n}(1 + ⟨v_{n}, w_{n}⟩) and ⟨T z_{n}, u_{n}⟩ = γ_{n}(1 + ⟨v_{n}, w_{n}⟩) we obtain:

\overline{α_{n}}⟨T u_{n}, z_{n}⟩ + α_{n}⟨T z_{n}, u_{n}⟩ = (\overline{α_{n}}β_{n} + α_{n}γ_{n})(1 + ⟨v_{n}, w_{n}⟩) =
r_{n}(|β_{n}| + |γ_{n}|)η_{n}(1 + ⟨v_{n}, w_{n}⟩)

Hence, from (1.1) we deduce that

r_{n}(|β_{n}| + |γ_{n}|)| Im[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| ≤ |δ_{n}| + 4r_{n}^{2}∥T ∥

so, we obtain:

(|β_{n}| + |γ_{n}|)| Im[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| → 0. (1.4)

We claim that also

(|β_{n}| + |γ_{n}|)| Re[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| → 0. (1.5)

To prove this claim we define x_{n} := u_{n}+α_{n}z_{n} whenever Im[\overline{α_{n}}⟨T u_{n}, z_{n}⟩+α_{n}⟨T z_{n}, u_{n}⟩] ≥ 0 and x_{n} := u_{n} − α_{n}z_{n} otherwise. Now we consider two particular cases:

(i) In the case that Re⟨T x_{n}, x_{n}⟩ = 0 for all n it follows from (1.3) that r_{n}(|β_{n}| + |γ_{n}|)| Re[η_{n}(1+⟨v_{n}, w_{n}⟩)]| ≤ |δ_{n}|+4r_{n}^{2}∥T ∥. Thus (|β_{n}|+|γ_{n}|)| Re[η_{n}(1+⟨v_{n}, w_{n}⟩)]| → 0, as desired.

(ii) Assume now that Re⟨T x_{n}, x_{n}⟩ ≠ 0 for all n. Also Im⟨T x_{n}, x_{n}⟩ ≠ 0 because the curvature at λ = 0 is infinite so that the only real point of W (T) is zero. Furthermore, it follows that

μ_{n} := \frac{| Im⟨T x_{n}, x_{n}⟩|}{Re^{2}⟨T x_{n}, x_{n}⟩} → ∞,

since ⟨T x_{n}, x_{n}⟩ → 0 with ∥x_{n}∥ → 1 . Therefore by (1.2)

| Re⟨T x_{n}, x_{n}⟩| = (\frac{| Im⟨T x_{n}, x_{n}⟩|}{μ_{n}})^{\frac{1}{2}} ≤

(\frac{2(|δ_{n}| + 4r_{n}^{2}∥T ∥)}{μ_{n}})^{\frac{1}{2}} .

However by (1.3)

| Re⟨T x_{n}, x_{n}⟩| ≥ r_{n}(|β_{n}| + |γ_{n}|)| Re η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| − |δ_{n}| − 4r_{n}^{2}∥T ∥.

Hence,

(\frac{2(|δ_{n}| + 4r_{n}^{2}∥T ∥)}{μ_{n}})^{\frac{1}{2}} + |δ_{n}| + 4r_{n}^{2}∥T ∥

r_{n}(|β_{n}| + |γ_{n}|)| Re[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| ≤

and so (|β_{n}| + |γ_{n}|)| Re[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| → 0, as desired.

Thus, from (i) and (ii), the above claim follows. Therefore, by (1.4), we see that

(|β_{n}| + |γ_{n}|)|(1 + ⟨v_{n}, w_{n}⟩)| = (|β_{n}| + |γ_{n}|)|η_{n}(1 + ⟨v_{n}, w_{n}⟩)| =
(|β_{n}| + |γ_{n}|) (Re^{2}[η_{n}(1 + ⟨v_{n}, w_{n}⟩)] + Im^{2}[η_{n}(1 + ⟨v_{n}, w_{n}⟩)])^{\frac{1}{2}} → 0.

But |1 + ⟨v_{n}, w_{n}⟩| ≥ 1 because Re⟨v_{n}, w_{n}⟩ ≥ 0, for every n. Thus |β_{n}| + |γ_{n}| → 0 which means precisely that ∥T u_{n}∥ → 0 as well as ∥T^{∗}u_{n}∥ → 0, so that (a) is proved.

In order to prove (b), assume for example that zero is a point of infinite righthand curvature (the case of infinite lefthand curvature may be handled similarly). Let α_{0} be in W_{s}(T) such that Re α_{0} > 0. Also Im α_{0} > 0 since otherwise zero would not be a point of infinite righthand curvature. Let ε_{n} be a sequence of positive real numbers such that ε_{n} → 0 and let u_{n} be a sequence of unit vectors in H such that ⟨T u_{n}, u_{n}⟩ = ε_{n}α_{0}. As in the proof of (a), we write T u_{n} = δ_{n}u_{n} + β_{n}v_{n} and T^{∗}u_{n} = \overline{δ_{n}}u_{n} + \overline{γ}_{n}w_{n}, where v_{n} and w_{n} are as before. However, in this case we have the following additional information δ_{n} = ε_{n}α_{0}. Let z_{n} := v_{n} + w_{n} and M be a constant greater than \frac{|α_{0}|| Re⟨T z_{n},z_{n}⟩|}{Re α_{0}} (take for instance M = \frac{5|α_{0}|∥T ∥}{Re α_{0}}). We define again

α_{n} := r_{n}τ_{n}, for every n, where τ_{n} is as before but now r_{n} := (\frac{|δ_{n}|}{M})^{\frac{1}{2}} . Then,

|α_{n}|^{2}|⟨T z_{n}, z_{n}⟩| = ε_{n} \frac{|α_{0}|| Re⟨T z_{n}, z_{n}⟩|}{M} < ε_{n} Re α_{0} = Re⟨T u_{n}, u_{n}⟩

so that

Re⟨T u_{n}, u_{n}⟩ + |α_{n}|^{2} Re⟨T z_{n}, z_{n}⟩ > 0.

On the other hand, for every n, there exists a complex number η_{n} with |η_{n}| = 1 such that

\overline{α_{n}}⟨T u_{n}, z_{n}⟩ + α_{n}⟨T z_{n}, u_{n}⟩ = (\frac{|δ_{n}|}{M})^{\frac{1}{2}} (|β_{n}| + |γ_{n}|)η_{n}(1 + ⟨v_{n}, w_{n}⟩).

Now we define x_{n} := u_{n} + α_{n}z_{n} if Re η_{n}(1 + ⟨v_{n}, w_{n}⟩) > 0 and x_{n} := u_{n} − α_{n}z_{n} otherwise. Then, Re⟨T x_{n}, x_{n}⟩ > 0. Consequently, since there are not positive real numbers in W_{s}(T) we deduce that Im⟨T x_{n}, x_{n}⟩ ≠ 0 for all n. Also, again because zero is a point of infinite righthand curvature and ⟨T x_{n}, x_{n}⟩ → 0 with ∥x_{n}∥ → 1, it follows that \frac{| Im⟨T x_{n},x_{n}⟩|}{Re^{2}⟨T x_{n},x_{n}⟩} → ∞. Then, by proceding as above in the corresponding part of the proof of (a), we deduce that (|β_{n}| + |γ_{n}|)| Re[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| → 0. On the other hand, from (1.2) we obtain (|β_{n}| + |γ_{n}|)| Im[η_{n}(1 + ⟨v_{n}, w_{n}⟩)]| → 0. Therefore we conclude that (|β_{n}| + |γ_{n}|)|η_{n}(1 + ⟨v_{n}, w_{n}⟩)| → 0 and so |β_{n}| + |γ_{n}| → 0. This proves that ∥T u_{n}∥ → 0 and ∥T^{∗}u_{n}∥ → 0.

Assertion (a) in the next corollary constitutes an improvement of the main result in [6]. Also assertion (b) shows in particular that points of infinite one-sided curvature of ∂W_{s}(T) which are not of infinite curvature are reducing essential eigenvalues. Finally the assertion (c) improves [7] , Corollary 4.

Corollary 1.8. (a) If λ is either a point of infinite curvature or a point of infinite one-sided curvature of ∂W_{s}(T), then λ ∈ R(T).

(b) If λ is a point of infinite one-sided curvature in ∂W_{s}(T) but it is not a linear vertex, then λ ∈ R_{e}(T).

(c) If λ is a point of infinite curvature in ∂W_{s}(T), but it is not a linear vertex of W (T), then λ is a normal essential eigenvalue of T.

Proof. Part (a) is an immediate consequence of the the definition of R(T) and Theorem 1.7. In order to prove (b) and (c) we first claim that a point λ in W (T) which is not a linear vertex belongs to R_{e}(T). To show the claim we consider T decomposed in the (direct) sum T = A ⊕ B, acting on H = H_{0}^{⊥} ⊕ H_{0} where H_{0} := Ker(T −λI)∩Ker(T^{∗} −\overline{λ}I). Since A and B are the restriction of T to H_{0}^{⊥} and H_{0} respectively, it is clear that λ is not a reducing eigenvalue of A so, in particular λ ∉ R_{00}(A). Notice that it is easily proved that W (T) = Convhull({λ}, W (A)) (see also [8]). Hence λ must be either in ∂W (A) or in the complement of W_{s}(A). In the first case, W (T) = W (A). The second alternative is not possible since, in that case, λ would be a linear vertex of W_{s}(T) which contradicts our working assumption. Since λ is a point of infinite curvature in ∂W_{s}(A) we deduce, from (a), that λ ∈ R(A). Because λ ∉ R_{00}(A), we conclude that λ ∈ R_{e}(A). Consequently λ ∈ R_{e}(T) and the claim is shown. To complete the proof of the theorem we merely observe that if u_{n} is a orthonormal sequence such that ∥(T − λI)u_{n}∥ → 0 then ⟨T u_{n}, u_{n}⟩ → λ. Hence, as a consequence of Theorem 1.7, we see that also ∥(T − λ)^{∗}u_{n}∥ → 0.

The following result proves a strengthening of a conjecture of Mathias Hübner [6].

Corollary 1.9. Assume that λ is a point of infinite one-sided curvature of ∂W_{s}(T), for some operator T, and suppose that there is only one tangent line to ∂W_{s}(T) at the point λ. Then λ ∈ R_{e}(T). If, in addition, λ is a point of infinite curvature of W_{s}(T) then λ is a normal essential eigenvalue of T.

Proof. It follows immediately from part (c) of the previous corollary because λ can not be a linear vertex of ∂W_{s}(T).

2. Joint Reducing Essential Eigenvalues and the Joint Numerical Range

Now, we shall find a mild extension of the last corollary to n-tuples.

Definition 2.10. Let T = (T_{1}, T_{2}, …, T_{n}) be an n-tuple of operators acting on H. The joint numerical range and the joint state numerical range of T are defined, respectively, as follows:

W (T) := {(⟨T_{1}x, x⟩, ⟨T_{2}x, x⟩, …, ⟨T_{n}x, x⟩) : x ∈ H, ∥x∥ = 1},
W_{s}(T) := {(ϕT_{1}, ϕT_{2}, …, ϕT_{n}) : ϕ ∈ S(C^{∗}(T))},

where C^{∗}(T) denotes the C^{∗}-algebra generated by T and the identity operator I on H.

Remark 2.11. (a) W_{s}(T) is always convex but, in general, W_{s}(T) = \overline{W (T)} only for n = 1. Indeed, for n > 1 the above equality does not always hold and W (T) is not convex in general [3].

(b) If λ is an extreme point of ∂W_{s}(T), for a given n-tuple of operators T, then λ ∈ \overline{W (T)}. Indeed, the proof of this fact uses a standard argument: let Σ_{λ} be the subset of S(C^{∗}(T)) consisting of those states ϕ on C^{∗}(T) such that ϕ(T) = λ. Since Σ_{λ} is compact and convex in the w^{∗}-topology, there exists an extreme point ψ in Σ_{λ}. Using the fact that λ is an extreme point of W (T), it readily follows that ψ is a pure state of C^{∗}(T). By Glim’s Lemma (see [4] and [5]), we obtain that ψ is in the w^{∗}-closure of the set of vector states of C^{∗}(T). Thus, evaluating at T, we conclude that λ ∈ \overline{W (T)}, as desired.

(c) In the next theorem we use the notions of joint reducing approximate point spectrum and of joint reducing essential spectrum of an n-tuple of operators. These are the natural extension to n-tuples of the corresponding notions for single operators.

Theorem 2.12. Given an n-tuple T = (T_{1}, ⋯ T_{n}) of operators acting on H, assume that λ = (λ_{1}, ⋯, λ_{n}) ∈ ∂W_{a}(T) satisfies λ_{j} is a point of one-sided infinite curvature of ∂W_{s}(T_{j}), for 1 ≤ j ≤ n. Then λ ∈ R(T). If, furthermore, λ is not a linear vertex of ∂W (T), then λ ∈ R_{e}(T). If, in addition, λ_{j} is a point of infinite curvature of ∂W_{s}(T_{j}), for 1 ≤ j ≤ n, then λ is a joint normal essential eigenvalue for T.

Proof. The proof consits of a repeated application of Corollary 1.8, by projecting in each coordinate. We should point out that under the present hypotheses, λ is an extreme point of \overline{W (T)}.

Remark 2.13. (a) It would be interesting to weaken the hypotheses of Corollary 1.8. For example, is it possible to simply assume that λ is a point of upper-infinite curvature in ∂W_{s}(T) (in the sense that we replace lim by lim sup in the definition of infinite curvature)?

(b) We recall that the m-th matricial range of an operator T on H is the set of m × m matrices of the form ϕ(T), where ϕ : C^{∗}(T) → M_{m} is a unital completely positive linear map (see, for instance, [2]). Would it be possible to define a notion of points of infinite matricial curvature in the boundary of the matricial range of T, so that to be able to conclude that such points are actually in the approximate reducing matricial spectrum of T (those ϕ(T) for which ϕ is actually a *-homomorphism)?

References

1. J. Agler, Geometric and topological properties of the numerical range, Indiana Univ. Math. J. 31 (1982), 767-777.

2. J. Bunce & N. Salinas, Completely positive maps on C^{∗}-algebras and the left matricial spectra of an operator, Duke Math. J. 74 (1976), 747-773.

3. A. T. Dash, Joint numerical ranges, Glasnik Matematicki (1972) 75-81.

4. J. Dixmier, Les C^{∗}-algèbres at leurs represéntations, Gauthier-Villars, Paris (1964).

5. J. Glimm, A Stone-Weierstrass theorem for C^{∗}-algebras, Ann. Math. 72 (1960), 216-244.

6. M. Hübner, Spectrum where the boundary of the numerical range is not round, To appear in Rocky Mountain Journal of Math..

7. J. Lancaster, The boundary of the numerical range, Proc. Amer. Math. Soc. 49 (1975), 393-398.

8. M. Radjabalipour & H. Radjavi, On the geometry of numerical ranges, Pacific J. Math. 61 (1975), 507-511.

9. N. Salinas, Reducing essential eigenvalues, Duke. Math. J. 40 (1973), 561-580.

10. J. G. Stampfli & J.P. Willians, Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. Jour. 20 (1968), 417-424.

Norberto Salinas, Department of Mathematics, University of Kansas, Lawrence KS 66045

 E-mail address: norberto@@kuhub.cc.ukans.edu

Maria Victoria Velasco, Department of Analysis, University of Granada, Granada, Spain

 E-mail address: vvelasco@@goliat.ugr.es