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Weyl’s theorem for a generalized derivation and an elementary operator. (English) Zbl 1093.47509

From the author’s abstract: For \(a,b\in B(H)\), with \(B(H)\) denoting the algebra of operators on a complex infinite-dimensional Hilbert space \(H\), the generalized derivation \(\delta_{a,b}\in B(B(H))\) and the elementary operator \(\Delta_{a,b}\in B(B(H))\) are defined by \(\delta_{a,b}(x)=ax-xb\) and \(\Delta_{a,b}(x)=axb-x\). Let \(d_{a,b}=\delta_{a,b}\) or \(\Delta_{a,b}\). It is proved if \(a,b^*\) are hyponormal, then \(f(d_{a,b})\) satisfies the (generalized) Weyl theorem for each function \(f\) analytic on a neighbourhood of \(\sigma(d_{a,b})\).

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47B20 Subnormal operators, hyponormal operators, etc.
47A53 (Semi-) Fredholm operators; index theories
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