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Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain

Published online by Cambridge University Press:  20 November 2018

Lawrence A. Fialkow
Lawrence A. Fialkow*
Affiliation:
Western Michigan University, Kalamazoo, Michigan
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Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1205 - 1231
Copyright
Copyright © Canadian Mathematical Society 1981

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