Spectral mapping theorems for the upper Weyl and upper Browder spectra

Abstract

We discuss the spectral mapping properties of the upper Weyl and upper Browder spectra of an arbitrary ordered Banach algebra element.     

SVEP and Generalized Weyl’s Theorem

We use localised single-valued extension property to prove generalized Weyl/Browder and a-generalized Weyl/Browder type theorems for Banach space operators.     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

Ruston,Riesz and perturbation classes

We determine the perturbation classes of Fredholm and Weyl elements, as well the “commuting perturbation classes” of Fredholm, Weyl and Browder elements with respect to unbounded Banach algebra homomorphism T. Among other things we use the Ruston elements of Mouton, Mouton and Raubenheimer. Also, we investigate the class of polynomially almost T null and the class of polynomially T Riesz elements.     

The Weyl’s Theorem and Its Perturbations in Semisimple Banach Algebra

Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl's theorem and the Browder's theorem for a,and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl's theorem or the Browder's theorem,where f is a complex-valued function analytic on a neighborhood of σ(a).In addition,the perturbations of the Weyl's theorem and the Browder's theorem are investigated.     

Essential,Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.     

Gustafson,weidmann, kato,wolf, schechter and browder essential spectra of some matrix operator and application to two‐group transport equation

In this paper, we investigate the Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of a 2 × 2 block matrix operator defined on a Banach space where entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components under some properties. Furthermore, we give an application to two‐group transport equation. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

Paranormal and ( p, k )-quasihyponormal operators: Spectral continuity

Continuity of the set theoretic functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum on the classes consisting of (p, k)-quasihyponormal,M-hyponormal, totally *-paranormal and paranormal (Hilbert space) operators is proved.     

The index for Fredholm elements in a Banach algebra via a trace II

We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index.     

Spectral inclusions for operators on Banach spaces

This article centers around the relation between the spectra of two Banach space operators that are linked by some intertwining condition such as quasi-similarity. Certain conditions from local spectral theory are shown to be both necessary and sufficient for these operators to have equal spectra, approximate point spectra, or surjectivity spectra. A key role is played by a localized version of Bishop’s classical property (β) and a related closed range condition. As an application to harmonic analysis, the measures on a locally compact abelian group that avoid the Wiener-Pitt phenomenon are characterized in terms of local spectral theory.     

On a characterization of the essential spectra of the sum and the product of two operators

In this paper, we consider the sum and the product of two operators acting on a Banach space and we present some new and quite general conditions to investigate their Wolf, Schechter and Browder essential spectra.     

The upper Browder spectrum property

In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism \(T: A \rightarrow B,\) we have inclusions

$$\begin{aligned} \sigma (Ta) \subseteq \beta _T(a) \subseteq \beta _T^+(a) \subseteq \sigma (a), \end{aligned}$$

where \(\sigma (Ta),\) \(\beta _T(a),\) \( \beta _T^+(a)\) and \( \sigma (a)\) denote the Fredholm, Browder, upper Browder and (usual) spectra of a,  respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element?

     

Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces

The purpose of this paper is to study iterative schemes of Browder and Halpern types for a semigroup of nonexpansive mappings on a compact convex subset of a smooth (and strictly convex) Banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented.     

Essential spectra of operator matrices and applications

The main goal of this paper is to give a characterization of Wolf, Schechter, and Browder essential spectra of (2 × 2) matrix operator on a Banach space. Furthermore, we apply the obtained results to transport equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

An extension of A. Ostrowski's Theorem on the round-off stability of iterations

Summary A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.     

Some spectral mapping theorems through local spectral theory

The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors [14], [29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems, as well as, the spectral mapping theorem relative to some other classical spectra. We also prove that ifT orT* has the single-valued extension property some of the more important spectra originating from Fredholm theory coincide. This result is extended, always in the caseT orT* has the single valued extension property, tof(T), wheref is an analytic function defined on an open disc containing the spectrum ofT. In the last part we improve a recent result of Curto and Han [10] by proving that for every transaloid operatorT a-Weyl’s theorem holds forf(T) andf(T)*. The research was supported by the International Cooperation Project between the University of Palermo (Italy) and Conicit-Venezuela.     

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T^∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.