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Convexoid and generalized derivations
Linear Algebra and its Applications 350 (2002) 289–292 www.elsevier.com/locate/laa
Convexoid and generalized derivations M. Barraa Department of Mathematics, Faculty of Sciences, Semlalia, Marrakech, Morocco Received 30 May 2001; accepted 13 February 2002 Submitted by V. Mehrmann
Abstract Let E, F be two Banach spaces and let S be a symmetric norm ideal of L(E, F ). For A ∈ L(F ) and B ∈ L(E) the generalized derivation δS,A,B is the operator on S that sends X to AX − XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δS,A,B is convexoid if and only if A and B are convexoid. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Elementary operator; Convexoid operator; Numerical range spectrum; Derivation
1. Introduction Let E, F be two complex Banach spaces and let L(E, F ) denote the Banach space of all bounded linear operators from E to F. We abbreviate L(E, E) to L(E). For A ∈ L(E) and B ∈ L(F ), The left and right multiplications LA ∈ L(L(F, E)) and RB ∈ L(L(F, E)) are defined by LA (X) = AX and RB (X) = XB, respectively. The generalized derivation induced by A ∈ L(E) and B ∈ L(F ) is the operator: δA,B : L(F, E) → L(F, E), X → AX − XB = (LA − RB )(X). Let S be a (not necessarily closed) linear subspace of L(F, E), and suppose S = / 0. Following [6], we refer to S as a symmetric norm ideal if the following conditions are satisfied: (i) If A ∈ L(E), B ∈ L(F ) and X ∈ S then AX ∈ S and XB ∈ S. (ii) S is a Banach space with respect to a norm | · |. (iii) |X| = X for all X ∈ S with one-dimensional range. (iv) |AXB| A |X| B for all A ∈ L(F ), B ∈ L(E), X ∈ S. E-mail address: [email protected] (M. Barraa). 0024-3795/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5( 0 2) 0 0 3 1 1 - 7
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Examples of symmetric norm ideals are the Schatten p-ideals (1 p ∞) of Hilbert space operators; see [3]. Of cource, L(F, E) itself is also a symmetric norm ideal. If S is a symmetric norm ideal, then δA,B (S) ⊂ S. We define δS,A,B ∈ L(S) as the restriction of δA,B to S. In the case where S is the Schatten p-ideal, we denote δS,A,B by δp,A,B . The (algebraic) numerical range V (T ) of an operator T ∈ L(E) is defined by V (T ) = {(T ) : ∈ L(E)∗ and = (I ) = 1}. It is well known that V (T ) is a compact convex set and that σ (T ), the spectrum of T, is a subset of V (T ). We refer to [1] for the basic facts about numerical ranges. The operator T is called convexoid if V (T ) coincides with co(σ (T )), the convex hull of σ (T ). The numerical range of generalized derivations on symmetric norm ideals was studied by several authors, including Kyle [5] and Shaw [8]. In [7], Seddik proved that δp,A,B is convexoid if and only if A and B are convexoid. We here extend this result to δS,A,B , that is, to the setting of arbitrary symmetric norm ideals of Banach spaces. Our approach is based on the formula: V (δS,A,B ) = V (A) − V (B), which was established by Mattila [6].
2. Spectra of generalized derivations For f ∈ F ∗ and x ∈ E we denote by x ⊗ f ∈ L(F, E) the operator that sends y ∈ F to f (y)x ∈ E. Lemma 2.1. Let A ∈ L(E), and B ∈ L(F ), then (i) σ (LS,A ) = σ (A) and σ (RS,B ) = σ (B). (ii) σ (δS,A,B ) ⊆ σ (A) − σ (B). Proof. (i) We confine ourselves to the proof of the first equality. Clearly, it suffices to show that A is invertible if and only if LS,A is invertible. If A is invertible and C is the inverse, then LS,C is obviously the inverse of LS,A . Conversely, suppose that LS,A is invertible. If there is an x ∈ E such that Ax = 0, then LS,A (x ⊗ f ) = 0 for every f ∈ F ∗ . This implies that x ⊗ f = 0 for all f ∈ F ∗ , whence x = 0. Thus, A is injective. To prove that A is surjective, pick y ∈ E. Since S contains all rank-one operators, we have y ⊗ f ∈ S for every f ∈ F ∗ . As LS,A is invertible, we can find an X ∈ S such that LS,A (X) = AX = y ⊗ f . By the Hahn–Banach theorem, there exists an f ∈ F ∗ such that f (x) = 1 for some x ∈ F . It results that AXx = y, which shows that y is on the range of A.
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(ii) Since LS,A and RS,B are two commuting operators, the asserted inclusion follows from (i) and the well-known result (see [4, p. 42]): Let S, T be two Banach space operators such that ST = T S, then σ (S + T ) ⊆ σ (S) + σ (T ).
Let σπ (T ) denote the approximate point spectrum of T. Lemma 2.2. σπ (A) − σπ (B ∗ ) ⊆ σ (δS,A,B ). Proof. If 0 ∈ σπ (A) − σπ (B ∗ ), then there exists a λ ∈ σπ (A) ∩ σπ (B ∗ ). Consequently, there are xn ∈ E and fn ∈ F ∗ such that xn = 1, (A − λ)xn → 0, fn = 1, (B ∗ − λ)fn → 0. Put Xn = xn ⊗ fn . Then Xn = 1 and δS,A,B (Xn ) = (A − λ)Xn − Xn (B − λ) → 0. Hence 0 ∈ σπ (δS,A,B ). Finally, if λ ∈ σπ (A) − σπ (B ∗ ), then 0 ∈ σπ (A − λ) − σπ (B ∗ ), and from what was already proved we deduce that 0 ∈ σπ (δS,A−λ,B ) = σ (δS,A,B ) − λ. Corollary 2.3. co(σ (δS,A,B )) = co(σ (A)) − co(σ (B)). Proof. From Lemma 2.1(ii) we infer that: co(σ (δS,A,B )) ⊆ co(σ (A) − σ (B)) = co(σ (A)) − co(σ (B)). To get the reverse inclusion, notice first that Fr(σ (T )), the boundary (= frontier) of σ (T ), is always contained in σπ (T ) and that co(Fr(σ (T )) = co(σ (T )) for every Banach space operator T. This and Lemma 2.2 give: co(σ (A)) − co(σ (B)) = co(σ (A)) − co(σ (B ∗ ) = co(Fr(σ (A))) − co(Fr(σ (B ∗ ))) ⊆ co(σπ (A) − co(σπ (B ∗ )) = co(σπ (A) − σπ (B ∗ )) ⊆ co(σ (δS,A,B )).
We remark that if E = F = H is a complex Hilbert space, then Corollary 2.3 can be essentially sharpened: Fialkow [2] proved that in this case we actually have σ (δS,A,B ) = σ (A) − σ (B).
3. Generalized derivations and convexoid operators We need the following lemma: Lemma 3.1 [7]. Let M, N, K, and L be four convex compact subsets of the complex field C. If M + N = K + L and M ⊆ K, N ⊆ L, then M = K and N = L.
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Theorem 3.2. Let A ∈ L(E), B ∈ L(F ) and S a symmetric norm ideal of L(F, E). Then A and B are convexoid if and only if δS,A,B is convexoid. Proof. If A and B are convexoid, then V (A) = co(σ (A)) and V (B) = co(σ (B)). Hence V (A) − V (B) = co(σ (A)) − co(σ (B)). From Corollary 2.3, we infer that co(σ (δS,A,B )) = co(σ (A)) − co(σ (B)) = V (A) − V (B). And by Mattila’s formula we have co(σ (δS,A,B )) = V (δS,A,B ) that is δS,A,B is convexoid. Conversely, suppose that δS,A,B is convexoid: co(σ (δS,A,B )) = V (δS,A,B ) = V (A) − V (B). By Lemma 2.1 σ (δS,A,B )) ⊆ σ (A) − σ (B). So co(σ (δS,A,B )) ⊆ co(σ (A) − σ (B)) = co(σ (A) − co((B). Consequently V (A) − V (B) ⊆ co(σ (A)) − co(σ (B)). This and Lemma 3.1 give V (A) = co(σ (A)) and V (B) = co(σ (B)). The following corollary is a generalization of Theorem 3.1 of [7]: Corollary 3.3. If J is a symmetric norm ideal of L(H ) where H is a complex Hilbert space, then, δJ,A,B is convexoid if and only if A and B are convexoid.
Acknowledgements The author would like to thank the referee for helping him improve the text.
References [1] F.F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, London Math. Soc. Lecture Note Ser., vol. 2, Cambridge, 1973. [2] L.A. Fialkow, A note on norm ideals and the operator X → AX − XB, Israel J. Math. 32 (1979) 331–348. [3] I.G. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moskow,1965; English trans.: Trans. Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969. [4] D.A. Herrero, Approximation of Hilbert Space Operators, vol. 1, second edition, Pitman Research Notes in Math Series 224, Longman Scientific and Technical, New York, 1989. [5] J. Kyle, Numerical ranges of derivations, Proc. Edinburgh Math. Soc. 21 (1978) 33–39. [6] K. Mattila, Complex strict and uniform convexity and hyponormal operators, Math. Proc. Cambridge Philos. 96 (1984) 483–493. [7] A. Seddik, The numerical range of elementary operators II, Linear Algebras Appl. 338 (2001) 239– 244. [8] S.H. Shaw, On numerical ranges of generalized derivations and related properties, J. Austral. Math. Soc. Ser A. 36 (1984) 134–142.
Convexoid and generalized derivations M. Barraa Department of Mathematics, Faculty of Sciences, Semlalia, Marrakech, Morocco Received 30 May 2001; accepted 13 February 2002 Submitted by V. Mehrmann
Abstract Let E, F be two Banach spaces and let S be a symmetric norm ideal of L(E, F ). For A ∈ L(F ) and B ∈ L(E) the generalized derivation δS,A,B is the operator on S that sends X to AX − XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δS,A,B is convexoid if and only if A and B are convexoid. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Elementary operator; Convexoid operator; Numerical range spectrum; Derivation
1. Introduction Let E, F be two complex Banach spaces and let L(E, F ) denote the Banach space of all bounded linear operators from E to F. We abbreviate L(E, E) to L(E). For A ∈ L(E) and B ∈ L(F ), The left and right multiplications LA ∈ L(L(F, E)) and RB ∈ L(L(F, E)) are defined by LA (X) = AX and RB (X) = XB, respectively. The generalized derivation induced by A ∈ L(E) and B ∈ L(F ) is the operator: δA,B : L(F, E) → L(F, E), X → AX − XB = (LA − RB )(X). Let S be a (not necessarily closed) linear subspace of L(F, E), and suppose S = / 0. Following [6], we refer to S as a symmetric norm ideal if the following conditions are satisfied: (i) If A ∈ L(E), B ∈ L(F ) and X ∈ S then AX ∈ S and XB ∈ S. (ii) S is a Banach space with respect to a norm | · |. (iii) |X| = X for all X ∈ S with one-dimensional range. (iv) |AXB| A |X| B for all A ∈ L(F ), B ∈ L(E), X ∈ S. E-mail address: [email protected] (M. Barraa). 0024-3795/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5( 0 2) 0 0 3 1 1 - 7
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M. Barraa / Linear Algebra and its Applications 350 (2002) 289–292
Examples of symmetric norm ideals are the Schatten p-ideals (1 p ∞) of Hilbert space operators; see [3]. Of cource, L(F, E) itself is also a symmetric norm ideal. If S is a symmetric norm ideal, then δA,B (S) ⊂ S. We define δS,A,B ∈ L(S) as the restriction of δA,B to S. In the case where S is the Schatten p-ideal, we denote δS,A,B by δp,A,B . The (algebraic) numerical range V (T ) of an operator T ∈ L(E) is defined by V (T ) = {(T ) : ∈ L(E)∗ and = (I ) = 1}. It is well known that V (T ) is a compact convex set and that σ (T ), the spectrum of T, is a subset of V (T ). We refer to [1] for the basic facts about numerical ranges. The operator T is called convexoid if V (T ) coincides with co(σ (T )), the convex hull of σ (T ). The numerical range of generalized derivations on symmetric norm ideals was studied by several authors, including Kyle [5] and Shaw [8]. In [7], Seddik proved that δp,A,B is convexoid if and only if A and B are convexoid. We here extend this result to δS,A,B , that is, to the setting of arbitrary symmetric norm ideals of Banach spaces. Our approach is based on the formula: V (δS,A,B ) = V (A) − V (B), which was established by Mattila [6].
2. Spectra of generalized derivations For f ∈ F ∗ and x ∈ E we denote by x ⊗ f ∈ L(F, E) the operator that sends y ∈ F to f (y)x ∈ E. Lemma 2.1. Let A ∈ L(E), and B ∈ L(F ), then (i) σ (LS,A ) = σ (A) and σ (RS,B ) = σ (B). (ii) σ (δS,A,B ) ⊆ σ (A) − σ (B). Proof. (i) We confine ourselves to the proof of the first equality. Clearly, it suffices to show that A is invertible if and only if LS,A is invertible. If A is invertible and C is the inverse, then LS,C is obviously the inverse of LS,A . Conversely, suppose that LS,A is invertible. If there is an x ∈ E such that Ax = 0, then LS,A (x ⊗ f ) = 0 for every f ∈ F ∗ . This implies that x ⊗ f = 0 for all f ∈ F ∗ , whence x = 0. Thus, A is injective. To prove that A is surjective, pick y ∈ E. Since S contains all rank-one operators, we have y ⊗ f ∈ S for every f ∈ F ∗ . As LS,A is invertible, we can find an X ∈ S such that LS,A (X) = AX = y ⊗ f . By the Hahn–Banach theorem, there exists an f ∈ F ∗ such that f (x) = 1 for some x ∈ F . It results that AXx = y, which shows that y is on the range of A.
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(ii) Since LS,A and RS,B are two commuting operators, the asserted inclusion follows from (i) and the well-known result (see [4, p. 42]): Let S, T be two Banach space operators such that ST = T S, then σ (S + T ) ⊆ σ (S) + σ (T ).
Let σπ (T ) denote the approximate point spectrum of T. Lemma 2.2. σπ (A) − σπ (B ∗ ) ⊆ σ (δS,A,B ). Proof. If 0 ∈ σπ (A) − σπ (B ∗ ), then there exists a λ ∈ σπ (A) ∩ σπ (B ∗ ). Consequently, there are xn ∈ E and fn ∈ F ∗ such that xn = 1, (A − λ)xn → 0, fn = 1, (B ∗ − λ)fn → 0. Put Xn = xn ⊗ fn . Then Xn = 1 and δS,A,B (Xn ) = (A − λ)Xn − Xn (B − λ) → 0. Hence 0 ∈ σπ (δS,A,B ). Finally, if λ ∈ σπ (A) − σπ (B ∗ ), then 0 ∈ σπ (A − λ) − σπ (B ∗ ), and from what was already proved we deduce that 0 ∈ σπ (δS,A−λ,B ) = σ (δS,A,B ) − λ. Corollary 2.3. co(σ (δS,A,B )) = co(σ (A)) − co(σ (B)). Proof. From Lemma 2.1(ii) we infer that: co(σ (δS,A,B )) ⊆ co(σ (A) − σ (B)) = co(σ (A)) − co(σ (B)). To get the reverse inclusion, notice first that Fr(σ (T )), the boundary (= frontier) of σ (T ), is always contained in σπ (T ) and that co(Fr(σ (T )) = co(σ (T )) for every Banach space operator T. This and Lemma 2.2 give: co(σ (A)) − co(σ (B)) = co(σ (A)) − co(σ (B ∗ ) = co(Fr(σ (A))) − co(Fr(σ (B ∗ ))) ⊆ co(σπ (A) − co(σπ (B ∗ )) = co(σπ (A) − σπ (B ∗ )) ⊆ co(σ (δS,A,B )).
We remark that if E = F = H is a complex Hilbert space, then Corollary 2.3 can be essentially sharpened: Fialkow [2] proved that in this case we actually have σ (δS,A,B ) = σ (A) − σ (B).
3. Generalized derivations and convexoid operators We need the following lemma: Lemma 3.1 [7]. Let M, N, K, and L be four convex compact subsets of the complex field C. If M + N = K + L and M ⊆ K, N ⊆ L, then M = K and N = L.
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Theorem 3.2. Let A ∈ L(E), B ∈ L(F ) and S a symmetric norm ideal of L(F, E). Then A and B are convexoid if and only if δS,A,B is convexoid. Proof. If A and B are convexoid, then V (A) = co(σ (A)) and V (B) = co(σ (B)). Hence V (A) − V (B) = co(σ (A)) − co(σ (B)). From Corollary 2.3, we infer that co(σ (δS,A,B )) = co(σ (A)) − co(σ (B)) = V (A) − V (B). And by Mattila’s formula we have co(σ (δS,A,B )) = V (δS,A,B ) that is δS,A,B is convexoid. Conversely, suppose that δS,A,B is convexoid: co(σ (δS,A,B )) = V (δS,A,B ) = V (A) − V (B). By Lemma 2.1 σ (δS,A,B )) ⊆ σ (A) − σ (B). So co(σ (δS,A,B )) ⊆ co(σ (A) − σ (B)) = co(σ (A) − co((B). Consequently V (A) − V (B) ⊆ co(σ (A)) − co(σ (B)). This and Lemma 3.1 give V (A) = co(σ (A)) and V (B) = co(σ (B)). The following corollary is a generalization of Theorem 3.1 of [7]: Corollary 3.3. If J is a symmetric norm ideal of L(H ) where H is a complex Hilbert space, then, δJ,A,B is convexoid if and only if A and B are convexoid.
Acknowledgements The author would like to thank the referee for helping him improve the text.
References [1] F.F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, London Math. Soc. Lecture Note Ser., vol. 2, Cambridge, 1973. [2] L.A. Fialkow, A note on norm ideals and the operator X → AX − XB, Israel J. Math. 32 (1979) 331–348. [3] I.G. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moskow,1965; English trans.: Trans. Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969. [4] D.A. Herrero, Approximation of Hilbert Space Operators, vol. 1, second edition, Pitman Research Notes in Math Series 224, Longman Scientific and Technical, New York, 1989. [5] J. Kyle, Numerical ranges of derivations, Proc. Edinburgh Math. Soc. 21 (1978) 33–39. [6] K. Mattila, Complex strict and uniform convexity and hyponormal operators, Math. Proc. Cambridge Philos. 96 (1984) 483–493. [7] A. Seddik, The numerical range of elementary operators II, Linear Algebras Appl. 338 (2001) 239– 244. [8] S.H. Shaw, On numerical ranges of generalized derivations and related properties, J. Austral. Math. Soc. Ser A. 36 (1984) 134–142.