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

M. Amouch, H. Zguitti (2008)

Mathematica Bohemica

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.

A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

Sen Huang (1999)

Studia Mathematica

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function φ a ( t ) : = φ ( α t a ) t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum σ w * ( φ a ) is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define Ʌ φ a to be the union of all...

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties ( SBaw ) , ( SBab ) , ( SBw ) and ( SBb ) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property ( SBab ) , then S T has the property ( SBab ) if and only if σ SBF + - ( S T ) = σ SBF + - ( S ) σ SBF + - ( T ) , where σ SBF + - ( T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties ( SBaw ) , ( SBb ) and ( SBw ) with...

Abstract Weyl-type theorems

Mohammed Berkani (2016)

Mathematica Bohemica

In this paper, we give a new approach to the study of Weyl-type theorems. Precisely, we introduce the concepts of spectral valued and spectral partitioning functions. Using two natural order relations on the set of spectral valued functions, we reduce the question of relationship between Weyl-type theorems to the study of the set difference between the parts of the spectrum that are involved. This study solves completely the question of relationship between two spectral valued functions, comparable...

Ascent spectrum and essential ascent spectrum

O. Bel Hadj Fredj, M. Burgos, M. Oudghiri (2008)

Studia Mathematica

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if σ a s c e ( T + F ) = σ a s c e ( T ) for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued extension...

Back to RS-SR spectral theory

C. Benhida, E. H. Zerouali (2007)

Banach Center Publications

Let X, Y be Banach spaces, S: X → Y and R: Y → X be bounded operators. We investigate common spectral properties of RS and SR. We then apply the result obtained to extensions, Aluthge transforms and upper triangular operator matrices.

B-Fredholm and Drazin invertible operators through localized SVEP

M. Amouch, H. Zguitti (2011)

Mathematica Bohemica

Let X be a Banach space and T be a bounded linear operator on X . We denote by S ( T ) the set of all complex λ such that T does not have the single-valued extension property at λ . In this note we prove equality up to S ( T ) between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.

Chain-finite operators and locally chain-finite operators.

Teresa Bermúdez, Antonio Martinón (1999)

Extracta Mathematicae

The problem we are concerned with in this research announcement is the algebraic characterization of chain-finite operators (global case) and of locally chain-finite operators (local case).

Classes of operators satisfying a-Weyl's theorem

Pietro Aiena (2005)

Studia Mathematica

In this article Weyl’s theorem and a-Weyl’s theorem on Banach spaces are related to an important property which has a leading role in local spectral theory: the single-valued extension theory. We show that if T has SVEP then Weyl’s theorem and a-Weyl’s theorem for T* are equivalent, and analogously, if T* has SVEP then Weyl’s theorem and a-Weyl’s theorem for T are equivalent. From this result we deduce that a-Weyl’s theorem holds for classes of operators for which the quasi-nilpotent part H₀(λI...

Decomposable embeddings, complete trajectories, and invariant subspaces

Ralph deLaubenfels, Phóng Vũ (1996)

Studia Mathematica

We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.

Extended Weyl type theorems

M. Berkani, H. Zariouh (2009)

Mathematica Bohemica

An operator T acting on a Banach space X possesses property ( gw ) if σ a ( T ) σ SBF + - ( T ) = E ( T ) , where σ a ( T ) is the approximate point spectrum of T , σ SBF + - ( T ) is the essential semi-B-Fredholm spectrum of T and E ( T ) is the set of all isolated eigenvalues of T . In this paper we introduce and study two new properties ( b ) and ( gb ) in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if T is a bounded linear operator acting on a Banach space X , then...

Generalized a-Weyl's theorem and the single-valued extension property.

Mohamed Amouch (2006)

Extracta Mathematicae

Let T be a bounded linear operator acting on a Banach space X such that T or T* has the single-valued extension property (SVEP). We prove that the spectral mapping theorem holds for the semi-essential approximate point spectrum σSBF-+(T); and we show that generalized a-Browder's theorem holds for f(T) for every analytic function f defined on an open neighbourhood U of σ(T): Moreover, we give a necessary and sufficient condition for such T to obey generalized a-Weyl's theorem. An application is given...

Generalized Weyl's theorem and quasi-affinity

Pietro Aiena, Mohammed Berkani (2010)

Studia Mathematica

A bounded operator T ∈ L(X) acting on a Banach space X is said to satisfy generalized Weyl's theorem if the complement in the spectrum of the B-Weyl spectrum is the set of all eigenvalues which are isolated points of the spectrum. We prove that generalized Weyl's theorem holds for several classes of operators, extending previous results of Istrăţescu and Curto-Han. We also consider the preservation of generalized Weyl's theorem between two operators T ∈ L(X), S ∈ L(Y) intertwined or asymptotically...

Hereditarily normaloid operators.

Bhagwati Prashad Duggal (2005)

Extracta Mathematicae

A Banach space operator T belonging to B(X) is said to be hereditarily normaloid, T ∈ HN, if every part of T is normaloid; T ∈ HN is totally hereditarily normaloid, T ∈ THN, if every invertible part of T is also normaloid; and T ∈ CHN if either T ∈ THN or T - λI is in HN for every complex number λ. Class CHN is large; it contains a number of the commonly considered classes of operators. We study operators T ∈ CHN, and prove that the Riesz projection associated with a λ ∈ isoσ(T), T ∈ CHN ∩ B(H)...

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