Perturbation of semi-Browder operators and stability of Browder's essential defect and approximate point spectrum (original) (raw)

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.

Browder and Semi-Browder Operators and Perturbation Function

This paper is devoted to the investigation of the stability of closed densely de¯ned semi-Browder and Browder operators on Banach spaces. Our approach consists to introduce the concepts of a perturbation function and a coperturbation function in order to deduce the stability under strictly singular and cosingular operator perturbations. Further, our results are used to show the invariance of Browder's spectrum.

Characterization of Closed Densely Defined Semi-Browder Linear Operators

Complex Analysis and Operator Theory, 2012

In the present paper we characterize the closed densely defined semi-Browder operators through the Kato decomposition. Furthermore, we apply the obtained results to give a new characterization of Browder's essential defect spectrum and Browder's essential approximate point spectrum under finite rank operator perturbations.

Continuity of spectra and compact perturbations

Bulletin of the Korean Mathematical Society

In this note we give conditions for continuity of spectrum, approximative point spectrum and defect spectrum on the set , where and is the set of compact operators.

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.

Spectral Mapping Theorems and Perturbation Theorems for Browder's Essential Spectrum

Transactions of the American Mathematical Society, 1970

If T is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of T, ess (T) [1]. We derive below spectral mapping theorems and perturbation theorems for Browder's essential spectrum. If T is a bounded linear operator and fis a function analytic on a neighborhood of the spectrum of T, we prove that /(ess (r)) = ess (J(T)). If T is a closed, densely defined linear operator with nonempty resolvent set and / is a polynomial, the same theorem holds. For a closed, densely defined linear operator T and a bounded linear operator B which commutes with T, we prove that ess (T+ B) S ess (T) + ess (B) = {fi + v : ¡i Eess (T), v e ess (B)}. By making additional assumptions, we obtain an analogous theorem for B unbounded.

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

Perturbation of operators and approximation of spectrum

Proceedings - Mathematical Sciences, 2014

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x) n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x) n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x) n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.