Geodesic
Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain
Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1205-1231

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Fialkow, Lawrence A. Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain. Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1205-1231. doi: 10.4153/CJM-1981-091-4
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