The Joint Essential Numerical Range of Operators: Convexity and Related Results

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2009

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li William & Mary, [email protected]

Yiu-Tung Poon

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Recommended Citation Li, C. K., & Poon, Y. T. (2009). The joint essential numerical range of operators: convexity and related results. Studia Mathematica, 194, 91-104.

This Article is brought to you for free and open access by the Arts and Sciences at W&M ScholarWorks. It has been accepted for inclusion in Arts & Sciences Articles by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. STUDIA MATHEMATICA 194 (1) (2009)

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li (Williamsburg, VA) and Yiu-Tung Poon (Ames, IA)

Abstract. Let W (A) and We(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A1,...,Am) acting on an inﬁnite-dimensional Hilbert space. It is shown that We(A) is always convex and admits many equivalent formulations. In particular, for any ﬁxed i ∈ {1, . . . , m}, We(A) can be obtained as the intersection of all sets of the form

cl(W (A1,...,Ai+1,Ai + F,Ai+1,...,Am)), where F = F ∗ has ﬁnite rank. Moreover, the closure cl(W (A)) of W (A) is always star- shaped with the elements in We(A) as star centers. Although cl(W (A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d ∈/ cl(W (A)), there is a linear functional f such that f(d) > sup{f(a): a ∈ cl(W (A˜ ))}, where A˜ is obtained from A by perturbing one of the components Ai by a ﬁnite rank self-adjoint operator. Other results on W (A) and We(A) extending those on a single operator are obtained.

1. Introduction. Let B(H) denote the algebra of bounded linear operators acting on a complex Hilbert space H. The numerical range of A ∈ B(H) is deﬁned as W (A) = {hAx, xi : x ∈ H, hx, xi = 1}, which is useful in studying operators; see [10, 11, 22, 24] and [25, Chapter 1]. Let S(H) denote the set of self-adjoint operators in B(H). Since every A ∈ B(H) admits a decomposition A = A1 + iA2 with A1,A2 ∈ S(H), we can identify W (A) with 2 {(hA1x, xi, hA2x, xi): x ∈ H, hx, xi = 1} ⊆ R . m This leads to the joint numerical range of A = (A1,...,Am) ∈ S(H) , m W (A) = {(hA1x, xi,..., hAmx, xi): x ∈ H, hx, xi = 1} ⊆ R ,

2000 Mathematics Subject Classiﬁcation: 47A12, 47A13, 47A55. Key words and phrases: joint essential numerical range, self-adjoint operator, Hilbert space.

DOI: 10.4064/sm194-1-6 [91] c Instytut Matematyczny PAN, 2009 92 C. K. Li and Y. T. Poon which has been studied by many researchers in order to understand the joint behavior of several operators A1,...,Am. One may see [1, 5, 12, 14, 15, 16, 19, 23, 28, 31, 33, 35] and their references for the background and many applications of the joint numerical range. Let F(H) and K(H) be the sets of finite rank and compact operators in B(H). In the study of finite rank or compact perturbations of operators, researchers consider the joint essential numerical range of A ∈ S(H)m defined by \ m m We(A) = {cl(W (A + K)) : K = (K1,...,Km) ∈ K(H) ∩ S(H) }.

Here cl(S) denotes the closure of the set S. For m = 2, We(A) can be identiﬁed with the essential numerical range of A = A1 + iA2 ∈ B(H), deﬁned by \ We(A) = {cl(W (A + K)) : K ∈ K(H)}. One may see [2, 3, 6, 7, 13, 18, 20, 21, 26, 27, 30, 32, 36, 37] for many interesting results on We(A) and We(A). In theoretical studies as well as applications, it is desirable to deal with A such that W (A) or cl(W (A)) is convex. For example, if cl(W (A)) is convex, one can apply the separation theorem to show that 0 ∈/ cl(W (A)) if and m Pm only if there exist r > 0 and c = (c1, . . . , cm) ∈ R such that ( i=1 ciAi) > rIH. Unfortunately, cl(W (A)) is not always convex. Here are some results concerning the convexity of W (A) and cl(W (A)), and related to We(A) (for example, see [5, 10, 11, 36, 21, 29, 31] and their references).

(P1) [31] W (A1,...,Am) is convex if

(a) span{I,A1,...,Am} has dimension at most 3, or (b) dim H ≥ 3 and span{I,A1,...,Am} has dimension at most 4.

(P2) [31] For any A1,A2,A3 ∈ S(H) such that span{I,A1,A2,A3} has dimension 4, there is always an A4 ∈ S(H) for which W (A1,...,A4) is not convex. (P3) [31] If m ≥ 4 then there exists A ∈ S(H)m such that W (A) is non-convex. m (P4) For any positive integer m and any A ∈ S(H) , We(A) is a compact set contained in W (A). If span{I,A1,...,Am} has dimension at most 4, then We(A) is convex. m (P5) [36] For S ⊆ R , let Ext(S) be the set of all points in S that do not lie in the open line segment joining two distinct points in S. Then Ext(cl(W (A))) ⊆ Ext(W (A)) ∪ Ext(We(A)). We remark that (P1)–(P3) also hold if we replace W (A) by cl(W (A)). In view of (P2) and (P3), if m > 3, then for A ∈ S(H)m and K ∈ m m K(H) ∩ S(H) the set cl(W (A + K)) is usually non-convex. Since We(A) Joint essential numerical range 93 is the intersection of non-convex sets, one does not expect the set We(A) to be convex. This might be the reason why the convexity of We(A) is seldom discussed for m > 3. In fact, some researchers have studied diﬀerent geo- metrical properties of We(A) under the assumption that We(A) is convex, and some have examined We(A) for diﬀerent classes of operators without discussing their convexity; for example, see [6, 26, 27, 30, 32].

In this paper, we prove the rather unexpected result that We(A) is always convex. Moreover, it is shown that the closure cl(W (A)) of W (A) is always star-shaped with the elements in We(A) as star centers. Many results relating We(A) and W (A) are also obtained. Our paper is organized as follows. In Section 2, we extend the results of [21] by establishing several equivalent formulations of the essential joint numerical range for A ∈ S(H)m. One key obstacle for such an extension is the fact that W (A) may not be convex. To get around this problem, we show that cl(W (A)) is star-shaped. The star-shapedness of cl(W (A)) and the conditions equivalent to membership in We(A), given in Section 2, lead to our main result that We(A) is convex and its elements are star centers of the set cl(W (A)), which is presented in Section 3. With the convexity theorem, we obtain additional descriptions of We(A) in Section 4 in terms of the perturbations of one of the components of A, and also in terms of linear combinations of the components of A. For example, we show that We(A1,...,Am) is equal to the sets \ {cl(W (A1,...,Ai−1,Ai + F,Ai+1,...,Am): F ∈ F(H) ∩ S(H)} and m m n X X o (a1, . . . , am): cjaj ∈ We cjAj for all (c1, . . . , cm) ∈ Ω , j=1 j=1 m Pm 2 where Ω = {(c1, . . . , cm) ∈ R : j=1 cj = 1}. Also, we obtain an analog of the separation theorem for the not necessarily convex set cl(W (A)), namely, for any element d ∈/ cl(W (A)), there is a linear functional f such that f(d) > sup{f(a): a ∈ cl(W (A˜ ))}, where A˜ is obtained from A by perturbing one of the components Aj by a ﬁnite rank self-adjoint operator. In Section 5, we present additional results on W (A) and We(A). For in- stance, We(A) = cl(W (A)) if and only if the extreme points of W (A) are contained in We(A); the convex hull of cl(W (A)) can always be realized as the joint essential numerical range of (A˜1,..., A˜m) for linear operators A˜1,..., A˜m acting on a separable Hilbert space. In our discussion, we always assume that H is inﬁnite-dimensional. For m any vector x ∈ H and A = (A1,...,Am) ∈ S(H) , we will use the notation

hAx, xi = (hA1x, xi,..., hAmx, xi). 94 C. K. Li and Y. T. Poon

m Furthermore, R will be used to denote the inner product space of 1 × m real vectors with the usual inner product hx, yi.

2. Equivalent conditions for We(A). Following [21, Theorem 5.1] and its corollary on a single operator A ∈ B(H), we obtain several conditions equivalent to membership in We(A). m Theorem 2.1. Let A = (A1,...,Am) ∈ S(H) . The following conditions are equivalent for a real vector a = (a1, . . . , am): T m m (1) a ∈ We(A) = {cl(W (A + K)) : K ∈ K(H) ∩ S(H) }. (2) a ∈ T{cl(W (A + F)) : F ∈ F(H)m ∩ S(H)m}. ∞ (3) There is an orthonormal sequence {xn}n=1 ⊂ H of vectors such that

lim hAxn, xni = a. n→∞ ∞ (4) There is a sequence {xn}n=1 ⊂ H of unit vectors converging weakly to 0 in H such that

lim hAxn, xni = a. n→∞ (5) There is an inﬁnite-dimensional projection P ∈ S(H) such that

P (Aj − ajI)P ∈ K(H) for j = 1, . . . , k. Most of the argument in [21] can be applied here except for one crucial step, where the convexity of W (A) for m = 2 is needed. Since W (A) may not be convex for m > 3, we need the following auxiliary result to overcome the obstacle. As a byproduct, it shows that cl(W (A)) is star-shaped. Theorem 2.2. Let A satisfy the hypothesis of Theorem 2.1, and let W3(A) be the set of real vectors a satisfying condition (3) of Theorem 2.1. Then W3(A) is non-empty and closed. Moreover, each element a ∈ W3(A) is a star center of cl(W (A)), i.e., for any b ∈ cl(W (A)) we have (1−t)a+tb ∈ cl(W (A)) for all 0 ≤ t ≤ 1. ∞ Proof. To prove that W3(A) is non-empty, let {xn}n=1 be an orthonor- ∞ mal sequence of vectors in H. Then the sequence {hAxn, xni}n=1 is bounded. By choosing a subsequence if necessary, we can assume that hAxn, xni converges. Hence, W3(A) is non-empty. Next, we show that W3(A) is closed. Suppose a ∈ cl(W3(A)). Then for n ∞ each n ≥ 1, there exists an orthonormal sequence {xk }k=1 such that n n n m n lim hAxk , xk i = a ∈ R and lim a = a. k→∞ n→∞ 2 Let δn = 1/(4n ). By going to subsequences if necessary, we may assume n n 2 that khAxk , xk i − ak < δn for all n, k. We may also assume that kA1k + 2 ··· + kAmk ≤ 1. Then khAx, yik ≤ kxk kyk for all x, y ∈ H. Joint essential numerical range 95

1 Choose x1 = x1. Then khAx1, x1i − ak < 1. Suppose we have chosen {x1,..., xn} orthonormal with khAxk, xki − ak < 1/k for 1 ≤ k ≤ n. Then choose N such that for all 1 ≤ k ≤ n, n+1 n+1 |hxk, xN i|, khAxk, xN ik < δn+1. n+1 Pn n+1 Let y = xN − k=1 hxN , xkixk. Then n+1 ky − xN k ≤ nδn+1, so 1 − nδn+1 ≤ kyk ≤ 1 + nδn+1. Therefore, khAy, yi − ak n+1 n+1 n+1 n+1 n+1 ≤ khA(y − xN ), yik + khAxN , y − xN ik + khAxN , xN i − ak n+1 n+1 ≤ ky − xN k(kyk + kxN k) + δn+1 ≤ (2n + 2)δn+1.

Let xn+1 = y/kyk. Then

kxn+1 − yk = |1 − kyk | ≤ nδn+1.

Hence, {x1,..., xn, xn+1} is an orthonormal set and

khAxn+1, xn+1i − ak ≤ ky − xn+1k(kyk + kxn+1k) + (2n + 2)δn+1 ≤ (4n + 3)δn+1 < 1/(n + 1).

To prove the last assertion, let a ∈ W3(A) and b ∈ cl(W (A)). Suppose {xn} is an orthonormal sequence in H such that hAxn, xni → a. For 0 ≤ t ≤ 1, we are going to show that (1 − t)a + tb ∈ cl(W (A)). Given ε > 0, let y be a unit vector in H such that khAy, yi − bk < ε. Choose n such that iθ khAxn, xni − ak < ε and√ khAy, x√nik < ε. Choose θ ∈ R such that he y, xni iθ is imaginary. Let z = t e y + 1 − t xn Then √ √ iθ iθ hz, zi = thy, yi + (1 − t)hxn, xni + 2 t 1 − t(he y, xni + hxn, e yi) = 1 and khAz, zi−((1 − t)a+tb)k ≤ (1−t)khAx , x i−ak+tkhAy, yi−bk √ √ n n iθ iθ + t 1 − tkhe Ay, xni+hAxn, e yik ≤ 2ε. Therefore, (1 − t)a + tb ∈ cl(W (A)).

The referee indicated that W3(A) is clearly closed, and a short proof is possible. We include a detailed proof for the sake of completeness and easy reference.

Proof of Theorem 2.1. For j = 2, 3, 4, 5, let Wj(A) be the set of a satisfying condition (j). Clearly, we have

W5(A) ⊆ W3(A) ⊆ W4(A) ⊆ We(A) ⊆ W2(A).

Suppose a ∈ W2(A). We are going to show that a ∈ W5(A). Without loss of generality, we may assume a = 0. 96 C. K. Li and Y. T. Poon

Since 0 ∈ W2(A) ⊆ cl(W (A)), there exists a unit vector x1 ∈ H such that khAx1, x1ik < 1/2. Suppose we have an orthonormal set {x1,..., xn} n such that khAxn, xnik < 1/2 . Let Q be the orthogonal projection of H onto the subspace S spanned by x1,..., xn and let

B = ((I − Q)A1(I − Q)|S⊥ ,..., (I − Q)Am(I − Q)|S⊥ ).

Let b = (b1, . . . , bm) ∈ W3(B) and bIS = (b1IS, . . . , bmIS). Then for Q = I − Q, we have

bIS ⊕ B = (b1Q + QA1Q, . . . , bmQ + QAmQ) = A + F m m for some F ∈ F(H) ∩ S(H) . Therefore, 0 ∈ cl(W (bIS ⊕ B)). Hence, there exists a unit vector x ∈ H such that kh(A + F)x, xik < 1/2n+2. Let x = y +z, where y ∈ S and z ∈ S⊥. Then kyk2 +kzk2 = kxk2 = 1. If z = 0, then h(A + F)x, xi = b ∈ W3(B) ⊆ cl(W (B)). If z 6= 0, then by Theorem 2.2, we have h(A + F)x, xi = kyk2b + kzk2hB(z/kzk), z/kzki ∈ cl(W (B)). ⊥ So there exists a unit vector xn+1 ∈ S such that 1 kh(A + F)x, xi − hBx , x ik < , n+1 n+1 2n+2 and hence 1 khAx , x ik = khBx , x ik < , n+1 n+1 n+1 n+1 2n+1 because hFxn+1, xn+1i = 0. Inductively, we can choose an orthonormal se- ∞ quence {xn}n=1 such that 1 (1) khAx , x ik < for all n ≥ 1. n n 2n

Let n1 = 1. For every 1 ≤ i ≤ m, we have ∞ ∞ X 2 2 X 2 ∗ 2 |hAixn1 , xni| ≤ kAixn1 k and |hAixn, xn1 i| ≤ kAi xn1 k . n=1 n=1

Hence, there exists n2 > n1 such that ∞ ∞ X 1 X 1 |hA x , x i|2 < and |hA x , x i|2 < i n1 n 2 i n n1 2 n=n2 n=n2 for all 1 ≤ i ≤ m. Repeating this procedure, we get a strictly increasing ∞ sequence {nk}k=1 of positive integers such that for all 1 ≤ i ≤ m, we have ∞ ∞ X 1 X 1 (2) |hA x , x i|2 < and |hA x , x i|2 < . i nk n 2k i n nk 2k n=nk+1 n=nk+1 Joint essential numerical range 97

Formulas (1) and (2) imply that ∞ X 2 (3) |hAixnk , xnl i| < ∞. k,l=1 ∞ Let P be the orthogonal projection onto the subspace spanned by {xnk }k=1. Then it follows from (3) that PAiP is compact for all 1 ≤ i ≤ m.

3. Convexity and star-shapedness m Theorem 3.1. Let A ∈ S(H) . Then We(A) is a compact convex subset of cl(W (A)). Moreover, each element in We(A) is a star center of the star- shaped set cl(W (A)).

Proof. Because We(A) is the intersection of compact sets, it is compact. To prove the convexity, let a, b ∈ We(A) and 0 ≤ t ≤ 1. Then for every m m F ∈ F(H) ∩ S(H) , we have a ∈ We(A) = We(A + F) and b ∈ We(A) ⊆ cl(W (A + F)). So, by Theorem 2.2, we have ta + (1 − t)b ∈ cl(W (A + F)). Hence, \ m m ta + (1 − t)b ∈ {cl(W (A + F)) : F ∈ F(H) ∩ S(H) } = We(A). By Theorems 2.1 and 2.2, we have the last assertion.

Note that We(A) ∩ W (A) may be empty. For example, if A = diag(1, 1/2, 1/3,... ) 2 acts on ` , then We(A) = {0} and W (A) = (0, 1]. One may wonder whether a point a ∈ We(A) ∩ W (A) is a star center of W (A). This is not true, as shown by the example below. Moreover, the example shows that for m ≥ 4 there exists A ∈ S(H)m such that cl(W (A)) is convex whereas W (A) is not. Of course, this is impossible for m ≤ 3 as W (A) is always convex.

2 Example 3.2. Consider H = ` with canonical basis {en : n ≥ 1}. Let A = (A1,...,A4) with

A1 = diag(1, 0, 1/3, 1/4,...),A2 = diag(1, 0) ⊕ 0, ! ! 0 1 0 i A3 = ⊕ 0,A4 = ⊕ 0. 1 0 −i 0

Then (1, 1, 0, 0) ∈ W (A) and (0, 0, 0, 0) ∈ W (A)∩We(A), but (1/2, 1/2, 0, 0) ∈/ W (A). Hence, W (A) is not convex. However, cl(W (A)) is convex.

Proof. Note that (1, 1, 0, 0) = hAe1, e1i ∈ W (A) and

(0, 0, 0, 0) = hAe2, e2i = lim hAen, eni ∈ W (A) ∩ We(A). n→∞ 98 C. K. Li and Y. T. Poon

P To show that (1/2, 1/2, 0, 0) ∈/ W (A), consider a unit vector x = xjej P∞ 2 such that n=1 |xn| = 1. If hA1x, xi = hA2x, xi = 1/2, then ∞ 2 X 2 2 |x1| + |xn| /n = |x1| = 1/2. n=3 2 2 Thus, xn = 0 for all n ≥ 3 and |x1| = |x2| = 1/2. It then follows that (hA3x, xi, hA4x, xi) 6= (0, 0). This proves that (1/2, 1/2, 0, 0) ∈/ W (A). Hence, (0, 0, 0, 0) ∈ We(A)∩W (A) is not a star center of W (A), and W (A) is not convex. To see that cl(W (A)) is convex, note that 0 ∈ We(A). Thus, by Theorem 3.1, for every b ∈ cl(W (A)) we have t0 + (1 − t)b ∈ cl(W (A)) for any t ∈ [0, 1]. Let B = (B1,B2,B3,B4), where

B1 = diag(0, 1, 0),B2 = diag(0, 1, 0), ! ! 0 1 0 i B3 = [0] ⊕ ,B4 = [0] ⊕ , 1 0 −i 0 and C = (C1,C2,C3,C4), where C1 = diag(1/3, 1/4,...) ⊕ [0], C2 = C3 = C4 = diag(0, 0,... ) ⊕ [0]. Then it is easy to verify that 4 2 2 2 W (B) = {(r, r, s, t) ∈ R : 4(r − 1/2) + s + t ≤ 1} and W (C) = {(c, 0, 0, 0) : c ∈ [0, 1/3]} are both compact and convex. Hence, W (B ⊕ C) = conv(W (B) ∪ W (C)) is compact and convex and W (A) ⊆ W (B ⊕ C) ⇒ cl(W (A)) ⊆ W (B ⊕ C). On the other hand, B ⊕ C = [0] ⊕ A ⊕ [0]. Therefore, W (B ⊕ C) = {t0 + (1 − t)b : b ∈ W (A)} ⊆ cl(W (A)). So, cl(W (A)) = W (B ⊕ C) is convex.

m 4. Other descriptions of We(A). For c = (c1, . . . , cm) ∈ R and A = m Pm (A1,...,Am) ∈ S(H) , let c·A = i=1 ciAi. Using the convexity of We(A), we obtain additional conditions equivalent to membership in We(A) in terms of c · A ∈ S(H) so that the joint behavior of A1,...,Am can be understood from their linear combinations. For A ∈ S(H) and a positive integer k, let

λk(A) = inf{max σ(A + F ): F ∈ S(H) with rank(F ) < k}. m m Theorem 4.1. Let A ∈ S(H) and a = (a1, . . . , am) ∈ R . Then a ∈ We(A) if and only if any one (and hence all) of the following conditions holds: Joint essential numerical range 99

m (1) For every c ∈ R , c · a ∈ We(c · A). m T (2) For every c ∈ R , c · a ∈ {cl(W (c · A + F )) : F ∈ F(H) ∩ S(H)}. m ∞ (3) For every c ∈ R , there is an orthonormal sequence {xn}n=1 ⊂ H such that

lim hc · Axn, xni = c · a. n→∞ m ∞ (4) For every c ∈ R , there is a sequence {xn}n=1 ⊂ H of unit vectors ∞ such that {xn}n=1 converges weakly to 0 in H and

lim hc · Axn, xni = c · a. n→∞ m (5) For every c ∈ R , there is an inﬁnite-dimensional projection P ∈ S(H) such that P (c · A − c · aI)P ∈ K(H). m (6) For every c ∈ R and k ≥ 1, λk(c · A − c · aI) ≥ 0.

Proof. By the convexity of We(A), we can apply the separation theorem to Theorem 2.1 to show that a ∈ We(A) if and only if any one of the conditions (1) to (5) holds. m To prove the equivalence of condition (6), suppose a ∈ R . Without loss of generality, we may assume that a = 0. Suppose 0 satisﬁes condition (6). m Then for every c ∈ R and F ∈ F(H) ∩ S(H) with rank(F ) = k, we have

λ1(c·A+F ) ≥ λk+1(c·A) ≥ 0 and λ1(−(c·A+F )) ≥ λk+1(−c·A) ≥ 0. Hence, c · 0 = 0 ∈ cl(W (c · A + F )). Therefore, condition (2) is satisﬁed. m Conversely, if 0 does not satisfy condition (6), then there exist c ∈ R and k ≥ 1 such that λk(c · A) < 0. Thus there exists F ∈ F(H) ∩ S(H) such that c · A + F < 0 and 0 does not satisfy condition (2). Let A ∈ S(H)m. Although the set cl(W (A)) may not be convex if m ≥ 4, we have the following analog of the separation theorem for a convex set. m Theorem 4.2. Let A = (A1,...,Am) ∈ S(H) and d = (d1, . . . , dm) m ∈ R . Then d ∈/ We(A) if and only if any one (and hence all) of the following conditions holds: (a) There exists K ∈ K(H)m ∩ S(H)m such that d ∈/ cl(W (A + K)). (b) There exists F ∈ F(H)m ∩ S(H)m with d ∈/ conv(cl(W (A + F))). m (c) There exist F ∈ F(H) ∩ S(H), r > 0 and c = (c1, . . . , cm) ∈ R such that m X (4) ci(Ai − diI) + F > rIH. i=1

Proof. For simplicity, replace (A1,...,Am) by (A1 − d1I,...,Am − dmI) and assume that d = (0,..., 0). 100 C. K. Li and Y. T. Poon

(c)⇒(b). If (c) holds, we may perturb (c1, . . . , cm) so that cj 6= 0 for all j ∈ {1, . . . , m} and condition (4) still holds true. In particular, c1 6= 0. Then let F = (F/c1, 0,..., 0). We have c · a > r > 0 for all a ∈ W (A + F). Therefore, 0 ∈/ conv(cl(W (A + F))). Clearly, we have (b)⇒(a), which implies that 0 ∈/ We(A). Finally, suppose 0 ∈/ We(A). Then by Theorem 4.1(2), there exist a real vector c = (c1, . . . , cm) and F ∈ F(H) ∩ S(H) such that 0 = c · 0 ∈/ cl(W (c · A + F )). Since cl(W (c · A + F )) is a closed subinterval [s, t] of R, Pm we may assume that 0 < s ≤ t. Let r = s/2. Then ( i=1 ciAi) + F > rIH. Hence, (c) holds. m Let Ω = {c ∈ R : hc, ci = 1}. By Theorem 4.2, we have the following result showing that We(A) can be expressed as the intersection of half- spaces. m Corollary 4.3. Let A = (A1,...,Am) ∈ S(H) . Then \ m We(A) = {d ∈ R : hc, di ≤ max We(c · A)} c∈Ω m = {d ∈ R : hc, di ∈ We(c · A) for all c ∈ Ω}. T For A ∈ B(H), let σe(A) = {σ(A + K): K ∈ K(H)} be the essential spectrum of A. Then for A ∈ S(H), we have

We(A) = conv σe(A).

Thus, one may replace max We(c · A) by max σe(c · A) in Corollary 4.3. m Corollary 4.4. Let A = (A1,...,Am) ∈ S(H) . If d ∈/ cl(W (A)), then for any i ∈ {1, . . . , m} there exists F ∈ F(H) ∩ S(H) such that d 6∈ conv(cl(W (A˜ ))), where A˜ = (A1,...,Ai−1,Ai + F,Ai+1,...,Am).

Proof. If d ∈/ cl(W (A)), then d ∈/ We(A). The result readily follows from the arguments in the last paragraph in the proof of Theorem 4.2. It follows from Theorem 2.1 that the intersection of the non-convex sets cl(W (A + K)), which equals We(A), is a convex set. By Theorem 4.2 and Corollary 4.4, we see that one can replace cl(W (A + K)) by its convex hull in the intersection to obtain the same convex set We(A). It is known that m for any B = (B1,...,Bm) ∈ B(H) ,

conv(cl(W (B))) = {(f(B1), . . . , f(Bm)) : f ∈ Ξ}, where Ξ is the set of linear functionals f on B(H) satisfying 1 = f(I) = max{f(X): X ∈ B(H), kXk ≤ 1} (for example, see [10, 11]). So, it is easier to determine conv(cl(W (A + K))) than cl(W (A + K)). In fact, we have the following. Joint essential numerical range 101

Corollary 4.5. Let A ∈ S(H)m and i ∈ {1, . . . , m}. Then

\ i−1 m−i We(A) = {cl(W (A + F)) : F ∈ {0} × (F(H) ∩ S(H)) × {0} } \ = {conv(cl(W (A + F))) : F ∈ {0}i−1 × (F(H) ∩ S(H)) × {0}m−i}.

Proof. Let F ∈ {0}i−1 × (F(H) ∩ S(H)) × {0}m−i. Clearly,

We(A) ⊆ cl(W (A + F)) ⊆ conv(cl(W (A + F))). So, we may take the intersection of the second and third sets over all F ∈ {0}i−1 × (F(H) ∩ S(H)) × {0}m−i, and get an inclusion involving the three sets in the corollary. Finally, if d ∈/ We(A), then d will not belong to the third set by Corollary 4.4. So, the third set is a subset of We(A). Hence, the three sets in the corollary are equal.

5. Additional results. The following result shows that We(A) is un- changed under certain operations on A. m Theorem 5.1. Let A = (A1,...,Am) ∈ S(H) . ⊥ (a) Suppose H1 is a closed subspace of H such that H1 is ﬁnite-dimen- ∗ sional. If X : H1 → H is such that X X = IH1 , then ∗ ∗ We(A) = We(X A1X,...,X AmX).

(b) For each j ∈ {1, . . . , m}, suppose Pj : H → H is an orthogonal projection such that I − Pj has ﬁnite rank. Then

We(A) = We(P1A1P1,...,PmAmPm). Proof. Use Theorem 2.1.

We will establish some additional relationships between the sets We(A) and W (A). The next theorem generalizes the results of [29] and [14]. m Theorem 5.2. Let A ∈ S(H) . Then We(A) = cl(W (A)) if and only if Ext(W (A)) ⊆ We(A).

Proof. If We(A) = cl(W (A)), then

Ext(W (A)) ⊆ W (A) ⊆ We(A).

Conversely, if Ext(W (A)) ⊆ We(A), then by (P5),

Ext(cl(W (A))) ⊆ We(A). Hence,

cl(W (A)) ⊆ conv(Ext(cl(W (A)))) ⊆ conv(We(A)) = We(A).

Since We(A) ⊆ cl(W (A)), we have We(A) = cl(W (A)). 102 C. K. Li and Y. T. Poon

For k ≥ 1, let Ik denote the k × k identity matrix. Then for A = m (A1,...,Am) ∈ S(H) , we have m A ⊗ Ik = (A1 ⊗ Ik,...,Am ⊗ Ik) ∈ S(H ⊕ · · · ⊕ H) . | {z } k

Similarly, let I∞ denote the identity operator acting on `2. Then for A = m (A1,...,Am) ∈ S(H) , we have m A ⊗ I∞ = (A1 ⊗ I∞,...,Am ⊗ I∞) ∈ S(H ⊕ H ⊕ · · ·) . | {z } inﬁnitely many m Theorem√ 5.3. Let A = (A1,...,Am) ∈ S(H) . Then for any positive integer k > m − 1,

W (A ⊗ Ik) = conv(W (A)). Moreover, We(A ⊗ I∞) = cl(conv(W (A))). √ Proof. Suppose k > m−1. By the result in [34], every a ∈ conv(W (A)) can be written as a = Pk t hAx , x i for some unit vectors x ,..., x √ j=1 j √j j 1 k ∈ H. Thus, for x = ( t1 x1,..., tk xk) ∈ H ⊕ · · · ⊕ H, we have hA ⊗ Ikx, xi = a. Conversely, if a = hA⊗Ikx, xi ∈ W (A⊗Ik), one can decompose the unit vector x into k parts y1,..., yk according to the structure of H⊗Ik. Then k X 2 a = kyjk hAyj/kyjk, yj/kyjki ∈ conv(W (A)). j=1

If a ∈ cl(conv(W (A))), then there is a sequence {xn} of unit vectors in H such that hAxn, xni → a. Let x˜n = 0,..., 0, xn, 0,... ∈ H ⊕ H ⊕ · · · . | {z } n−1

Then {xñ} is an orthonormal sequence in H ⊕ H ⊕ · · · and hA ⊗ I∞xñ, xñi → a. Therefore, a ∈ We(A ⊗ I∞). Since ∞ [ We(A ⊗ I∞) ⊆ cl(W (A ⊗ I∞)) = cl W (A ⊗ Ik) ⊆ cl(conv(W (A))), k=1 we get the reverse inclusion. m Corollary 5.4. Let S be a compact convex subset of R . Then there are A, A˜ ∈ S(H)m with H = `2 such that W (A) is convex and

W (A) ⊆ S = cl(W (A)) = We(A˜ ). 2 Proof. For j = 1, . . . , m, let Aj = diag(a1j, a2j,... ) act on ` with the standard canonical basis {en : n ≥ 1} and be such that {(ai1, . . . , aim): Joint essential numerical range 103 i ≥ 1} is a dense subset of S. Then for A = (A1,...,Am) the set

W (A) = conv{(ai1, . . . , aim): i ≥ 1} is convex, and A˜ = A ⊗ I∞ satisﬁes the assertion by Theorem 5.3. Acknowledgements. Research of both authors was supported by NSF. Li was also supported by the William and Mary Plumeri Award; he is an honorary professor of the University of Hong Kong and an honorary professor of the Taiyuan University of Technology.

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Department of Mathematics Department of Mathematics The College of William and Mary Iowa State University Williamsburg, VA 23185, U.S.A. Ames, IA 50011, U.S.A. E-mail: [email protected] E-mail: [email protected]

Received May 1, 2008 Revised version February 8, 2009 (6348)