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On numerical ranges of generalized derivations and related properties

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Sen-Yen Shaw
Sen-Yen Shaw
Affiliation:
Department of Mathematics National Central University Chung-Li, Taiwan 320 Republic of, China
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Abstract

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This paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.

Keywords

algebra numerical rangederivationHermitian and normal operatorshyponormal operatornormed operator idealSchatten p-classC*-algebraHlibert-Schmidt operator.

MSC classification

Secondary: 47A12: Numerical range, numerical radius 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B47: Commutators, derivations, elementary operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 134 - 142
Copyright
Copyright © Australian Mathematical Society 1984

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