## Hokkaido Mathematical Journal

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

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

Muneo CHŌ, Slavisa DJORDJEVIĆ, and Bhaggy DUGGAL

#### 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 *T*_{p} of the operator are poles of the resolvent of *T*_{p}; *T* is hereditarly normaloid, *T* ∈ (¥cal{HN}), if every part *T*_{p} 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 *d*_{AB} ∈ *B*(*B*(¥cal{H})) stands for either of the elementary operators δ_{AB}(*X*) = *AX* - *XB* and Δ_{AB}(*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*(*d*_{AB}) satisfies Weyl's theorem and *f*(*d*_{AB})^{*} satisfies *a*-Weyl's theorem for every function *f* which is analytic on a neighbourhood of σ(*d*_{AB}). Finally, we characterize perturbations of *d*_{AB} by quasinilpotent and algebraic operators *A*, *B* ∈ *B*(¥cal{H}).

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 26 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1319595859

**Digital Object Identifier**

doi:10.14492/hokmj/1319595859

**Mathematical Reviews number (MathSciNet)**

MR2883494

**Zentralblatt MATH identifier**

1228.47036

**Subjects**

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.

**Keywords**

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

#### Citation

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