Hokkaido Mathematical Journal

CHŌ Muneo, DJORDJEVIĆ Slavisa, DUGGAL Bhaggy,

Bishop's property (β) and an elementary operator.

Hokkaido Mathematical Journal, 40 (2011) pp.337-356

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Abstract

A Banach space operator T ∈ B(¥cal{X}) is hereditarily polaroid, T ∈ (¥cal{HP}), if the isolated points of the spectrum of every part Tp of the operator are poles of the resolvent of Tp; T is hereditarly normaloid, T ∈ (¥cal{HN}), if every part Tp of T is normaloid. Let (¥cal{HNP}) denote the class of operators T ∈ B(¥cal{X}) such that T ∈ (¥cal{HP}) ∩ (¥cal{HN}). (¥cal{HNP}) operators such that the Berberian-Quigley extension T° of T is also in (¥cal{HNP}) satisfy Bishop's property (β). Given Hilbert space operators A, B* ∈ B(¥cal{H}), let dAB ∈ B(B(¥cal{H})) stands for either of the elementary operators δAB(X) = AX - XB and ΔAB(X) = AXB - X. If A, B* ∈ (¥cal{HP}) satisfy property (β), and the eigenspaces corresponding to distinct eigenvalues of A (resp., B*) are orthogonal, then f(dAB) satisfies Weyl's theorem and f(dAB)* satisfies a-Weyl's theorem for every function f which is analytic on a neighbourhood of σ(dAB). Finally, we characterize perturbations of dAB by quasinilpotent and algebraic operators A, B ∈ B(¥cal{H}).

MSC(Primary)47B47
MSC(Secondary)47B10, 47A10, 47B40
Uncontrolled KeywordsHilbert space, elementary operator, polaroid operator, SVEP, property (b), Browder's theorem, Weyl's theorem, perturbation