For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article (Property (gz) for bounded linear operators, (2013)) introduced and studied the property (gaz). In this study, through techniques using the local spectral theory of operators, we discover the sufficient conditions that allow the transfer of the property (gaz) from two tensor factors T and S to their tensor product T⊗S. The stability of the property (gaz) in the tensor product under perturbations is also investigated. The theory is exemplified by considering suitable classes of operators such as shift operators, convolution operators, and m-invertible contractions.
In [8], Sanabria et al. have shown that Weyl's theorem holds for operators satisfying property(Saw). In this paper, it is shown that its converse is not true with examples and the required condition is derived.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.