Hokkaido Mathematical Journal

Bishop's property (β) and an elementary operator

Muneo CHŌ, Slavisa DJORDJEVIĆ, and Bhaggy DUGGAL

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A Banach space operator TB(¥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 TB(¥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 dABB(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, BB(¥cal{H}).

Article information

Hokkaido Math. J., Volume 40, Number 3 (2011), 337-356.

First available in Project Euclid: 26 October 2011

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Zentralblatt MATH identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc. 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47A10: Spectrum, resolvent 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.

Hilbert space elementary operator polaroid operator SVEP property (b) Browder's theorem Weyl's theorem perturbation


CHŌ, Muneo; DJORDJEVIĆ, Slavisa; DUGGAL, Bhaggy. Bishop's property (β) and an elementary operator. Hokkaido Math. J. 40 (2011), no. 3, 337--356. doi:10.14492/hokmj/1319595859. https://projecteuclid.org/euclid.hokmj/1319595859

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