Property (ω) and quasi-class (A, k) operators
In this paper, we prove the following assertions: (i) If T is of quasi-class (A, k), then T is polaroid and reguloid; (ii) If T or T* is an algebraically of quasi-class (A, k) operator, then Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iii) If T* is an algebraically of quasi-class (A, k) operator, then a-Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iv) If T* is algebraically of quasi-class (A, k) then property (ω) holds for T.
Invertibility and Singularity for Bounded Linear Operators
This introduction to functional analysis focuses on the types of singularity that prevent an operator from being invertible. The presentation is based on the open mapping theorem, Hahn-Banach theorem, dual space construction, enlargement of normed space, and Liouville's theorem. Suitable for advanced undergraduate and graduate courses in functional analysis, this volume is also a valuable resource for researchers in Fredholm theory, Banach algebras, and multiparameter spectral theory. The treatment develops the theory of open and almost open operators between incomplete spaces. It builds the enlargement of a normed space and of a bounded operator and sets up an elementary algebraic framework for Fredholm theory. The approach extends from the definition of a normed space to the fringe of modern multiparameter spectral theory and concludes with a discussion of the varieties of joint spectrum. This edition contains a brief new Prologue by author Robin Harte as well as his lengthy new Epilogue, "Residual Quotients and the Taylor Spectrum." Dover republication of the edition published by Marcel Dekker, Inc., New York, 1988.
Some characterizations of operators satisfying a-Browder's theorem
Abstract We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C . Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λ I − T .