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The numerical range of elementary operators

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Abstract

For n-tuplesA=(A 1,...,A n ) andB=(B 1,...,B n ) of operators on a Hilbert spaceH, letR A,B denote the operator onL(H) defined by\(R_{A,B} (X) = \sum _{i = 1}^n A_i XB_i \). In this paper we prove that

$$co\left\{ {\sum\limits_{i = 1}^n {\alpha _i \beta _i :(\alpha _i , \ldots ,\alpha _n ) \in W(A),(\beta _i , \ldots ,\beta _n ) \in W(B)} } \right\}^ - \subset W_0 (R_{A,B} )$$

whereW is the joint spatial numerical range andW 0 is the numerical range. We will show also that this inclusion becomes an equality whenR A,B is taken to be a generalized derivation, and it is strict whenR A,B is taken to be an elementary multiplication operator induced by non scalar self-adjoints operators.

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Authors and Affiliations

  1. Department of Mathematics, University of Sana'a Faculty of Science, P.O. Box 14026, Sana'a, Yemen

    A. Seddik

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  1. A. Seddik
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Seddik, A. The numerical range of elementary operators. Integr equ oper theory 43, 248–252 (2002). https://doi.org/10.1007/BF01200256

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  • DOI: https://doi.org/10.1007/BF01200256

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