A Note on Property (gb) and Perturbations. (21st August 2012)

• Record Type: Journal Article
• Authors: Zeng, Qingping; Zhong, Huaijie
• Other Names: Zelik Sergey V. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2012(2012)
• Publication Date: 2012-08-21
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2012/523986
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : An operator T ∈ ℬ ( X ) defined on a Banach space X satisfies property ( g b ) if the complement in the approximate point spectrum σ a ( T ) of the upper semi-B-Weyl spectrum σ S B F + - ( T ) coincides with the set Π ( T ) of all poles of the resolvent of T . In this paper, we continue to study property ( g b ) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with T . Two counterexamples show that property ( g b ) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.