On subclasses of Browder and Weyl operators (original) (raw)

On a new class of operators and Weyl type theorems

Filomat, 2013

In the present article, we introduce a new class of operators which will be called the class of k-quasi *-paranormal operators that includes '-paranormal operators. A part from other results, we show that following results hold for a k-quasi *-paranormal operator T: (i) T has the SVEP. (ii) Every non-zero isolated point in the spectrum of T is a simple pole of the resolvent of T. (iii) All Weyl type theorems hold for T. (iv) Comments and some open problems are also presented.

On Closed Upper and Lower Semi-Browder Operators

Mediterranean Journal of Mathematics, 2014

We give several necessary and sufficient conditions for a closed operator to be upper (lower) semi-Browder. We also apply these results to give some characterizations of upper (lower) semi-Browder spectrum.

Generalized Weyl's theorem for some classes of operators

Kyungpook Mathematical Journal, 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set σBw(A) of all λ ∈ C such that A−λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem σ Bw (A) = σ(A) \ E(A), and the B-Weyl spectrum σ Bw (A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in , if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

A note on the a-Browder’s and a-Weyl’s theorems

Let T be a Banach space operator. In this paper, we characterize a-Browder’s theorem for T by the localized single valued extension property. Also, we characterize a-Weyl’s theorem under the condition E a (T)=π a (T), where E a (T) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π a (T) is the set of all left poles of T. Some applications are also given.

New decompositions for the classes of quasi-Fredholm and semi B-Weyl operators

Linear and Multilinear Algebra, 2018

In this paper, we show that if T ∈ L(X) is a quasi-Fredholm operator of degree d such that N(T d) + R(T) is complemented, then T has a decomposition like the Kato type operators. Using this decomposition allows us get a result about the stability of this class of operators under perturbations by nilpotent operators. Also we give a new decomposition for the class of semi B-Weyl operators, and through this property we shown that T is a semi B-Weyl operator if and only if T = S+K where S is a semi B-Browder operator and K is finite-dimensional.

Browder and semi-browder operators

Acta Mathematica Scientia, 2012

In this article, we study characterization, stability, and spectral mapping theorem for Browder's essential spectrum, Browder's essential defect spectrum and Browder's essential approximate point spectrum of closed densely defined linear operators on Banach spaces.

On Left and Right Browder Operators

Journal of the Korean Mathematical Society, 2011

We discuss the perturbation theory of "left" and "right" Browder operators, which come somewhere between Browder operators and semi Browder operators.

Generalized Browder's and Weyl's theorems for Banach space operators

Journal of Mathematical Analysis and Applications, 2007

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T . We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M -hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f (T ), where f ∈ H((T )), the space of functions analytic on an open neighborhood of σ(T ). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f (T ), for each f ∈ H(σ(T )).

Some characterizations of operators satisfying a-Browder's theorem

Journal of Mathematical Analysis and Applications, 2005

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 λI − T . 531 denote the class of all upper semi-Fredholm operators, and let Φ − (X) := T ∈ L(X): β(T ) < ∞ denote the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by