# An elementary operator and generalized Weyl’s theorem

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Journal of Inequalities and Applications

, 2012:243

First Online: 24 October 2012Received: 10 April 2012Accepted: 09 October 2012

Abstract

A Hilbert space operator T belongs to class A if | T 2 | − | T | 2 ≥ 0 Open image in new window. Let d A B Open image in new window denote either δ A B Open image in new window or △ A B Open image in new window, where δ A B Open image in new window and △ A B Open image in new window denote the generalized derivation and the elementary operator on a Banach space B H Open image in new window defined by δ A B X = A X − X B Open image in new window and △ A B X = A X B − X Open image in new window respectively. If A and B ∗ Open image in new window are class A operators, we show that d A B Open image in new window is polaroid and generalized Weyl’s theorem holds for f d A B Open image in new window, generalized a-Weyl’s theorem holds for f d A B ∗ Open image in new window for every f ∈ H σ d A B Open image in new window and f is not constant on each connected component of the open set U containing σ d A B Open image in new window, where H σ d A B Open image in new window denotes the set of all analytic functions in a neighborhood of σ d A B Open image in new window.

MSC: 47B20, 47A63.

Keywordsclass A operators generalized derivation elementary operator generalized Weyl’s theorem generalized a-Weyl’s theorem Download fulltext PDF

Author: **Fugen Gao - Xiaochun Li**

Source: https://link.springer.com/