Table of Contents
ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 415980, 10 pages
http://dx.doi.org/10.5402/2011/415980
Research Article

Spectrum of Quasi-Class (𝐴,π‘˜) Operators

1College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China
2Department of Mathematics, Tongji University, Shanghai 200092, China

Received 21 January 2011; Accepted 28 March 2011

Academic Editor: W. Kryszewski

Copyright Β© 2011 Xiaochun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An operator π‘‡βˆˆπ΅(β„‹) is called quasi-class (𝐴,π‘˜) if π‘‡βˆ—π‘˜(|𝑇2|βˆ’|𝑇|2)π‘‡π‘˜β‰₯0 for a positive integer π‘˜, which is a common generalization of class A. In this paper, firstly we consider some spectral properties of quasi-class (𝐴,π‘˜) operators; it is shown that if 𝑇 is a quasi-class (𝐴,π‘˜) operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of 𝑇 are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class A operators. Particularly, we show that if 𝑇 is a class A operator and either 𝜎(|𝑇|) or 𝜎(|π‘‡βˆ—|) is not connected, then 𝑇 has a nontrivial invariant subspace.