0000000000057989
AUTHOR
Pietro Aiena
showing 84 related works from this author
Weyl's theorem for some classes of operators
2005
Weyl's Theorems and Extensions of Bounded Linear Operators
2012
A bounded operator $T\in L(X)$, $X$ a Banach space, is said to satisfy Weyl's theorem if the set of all spectral points that do not belong to the Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues and having finite multiplicity. In this article we give sufficient conditions for which Weyl's theorem for an extension $\overline T$ of $T$ (respectively, for $T$) entails that Weyl's theorem holds for $T$ (respectively, for $\overline T$).
Weyl's theorems and Kato spectrum
2007
Generalized Weyl's theorem and quasi-affinity
2010
Local Spectral Theory
2018
In this chapter we shall introduce an important property, defined for bounded linear operators on complex Banach spaces, the so-called single-valued extension property (SVEP).
Fredholm Spectra and Weyl Type Theorems for Drazin Invertible Operators
2016
In this paper we investigate the relationship between some spectra originating from Fredholm theory of a Drazin invertible operator and its Drazin inverse, if this does exist. Moreover, we study the transmission of Weyl type theorems from a Drazin invertible operator R, to its Drazin inverse S.
Property (gab) through localized SVEP
2015
In this article we study the property (gab) for a bounded linear operator T 2 L(X) on a Banach space X which is a stronger variant of Browder's theorem. We shall give several characterizations of property (gab). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gab) holds for large classes of operators and prove the stability of property (gab) under some commuting perturbations. 2010 Mathematics Subject Classication. Primary 47A10, 47A11; Secondary 47A53, 47A55.
Some perturbation results through localized SVEP
2016
Some classical perturbation results on Fredholm theory are proved and extended by using the stability of the localized single-valued extension property under Riesz commuting perturbations. In the last part, we give some results concerning the stability of property (gR) and property (gb.
Fredholm and local spectral theory, with applications to multipliers
2004
This book shows the deep interaction between two important theories: Fredholm and local spectral theory. A particular emphasis is placed on the applications to multipliers and in particular to convolution operators. The book also presents some important progress, made in recent years, in the study of perturbation theory for classes of operators which occur in Fredholm theory
On the stability of the localized single-valued extension property under commuting perturbations
2013
This article concerns the permanence of the single-valued extension property at a point under suitable perturbations. While this property is, in general, not preserved under sums and products of commuting operators, we obtain positive results in the case of commuting perturbations that are quasi-nilpotent, algebraic, or Riesz operators.
On the Dunford property (C) for bounded linear operators SR and RS
2011
In this paper we show that if S in L(X; Y ) and R in L(Y;X), X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C)
Weyl's and Browder's theorems through the quasi-nilpotent part of an operator
2006
Polaroid type operators under quasi-affinies
2010
We study the permanence of polaroid type conditions under quasi-affinities
Semi-Fredholm operators, perturbation theory and localized SVEP
2007
Tensor products, multiplications and Weyl's theorem
2005
Tensor products, multiplications and Weyl’s theorem
2005
Tensor productsZ=T 1⊗T 2 and multiplicationsZ=L T 1 R T 2 do not inherit Weyl’s theorem from Weyl’s theorem forT 1 andT 2. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ w(Ti) ofT i, and if the operatorsT i are Kato type at the isolated points ofσ(Ti), thenZ andZ* satisfy Weyl’s theorem.
On left and right poles of the resolvent
2008
Some characterizations of operators satisfying a-Browder's theorem
2005
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 .
Polaroid type operators under perturbations
2013
On Commuting Quasi-Nilpotent Operators that are Injective
2022
Banach space operators that commute with an injective quasi-nilpotent operator, 11 such as the Volterra operator, inherit spectral and Fredholm properties, relating in 12 particular to the Weyl spectra.
Classes of operators satisfying a-Weyl's theorem
2005
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(I T ) is equal to ker (I T ) p for some p2N and every 2C, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghi…
Single-valued extension property at the points of the approximate point spectrum
2003
Abstract A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ o ∈ C in the case that λoI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.
Some spectral mapping theorems through local spectral theory
2004
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 re…
Property (R) for Bounded Linear Operators
2011
We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid, or left polaroid, operators.
Essential Spectra Under Perturbations
2018
The spectrum of a bounded linear operator on a Banach space X can be sectioned into subsets in many different ways, depending on the purpose of the inquiry.
Browder-Type Theorems
2018
This chapter may be viewed as the part of the book in which the interaction between local spectral theory and Fredholm theory comes into focus. The greater part of the chapter addresses some classes of operators on Banach spaces that have a very special spectral structure. We have seen that the Weyl spectrum σw(T) is a subset of the Browder spectrum σb(T) and this inclusion may be proper. In this chapter we investigate the class of operators on complex infinite-dimensional Banach spaces for which the Weyl spectrum and the Browder spectrum coincide. These operators are said to satisfy Browder’s theorem. The operators which satisfy Browder’s theorem have a very special spectral structure, ind…
Weyl's type theorems for polaroid operators
2008
Teoremi di Weyl per operatori polaroidi
Local Spectral Properties Under Conjugations
2021
AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.
Operators Which Do Not Have the Single Valued Extension Property
2000
Abstract In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T ∈ L ( X ) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ o ∈ C .
Property (R) under perturbations
2012
Property (R) holds for a bounded linear operator $${T \in L(X)}$$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.
CHARACTERIZATIONS OF STRICTLY SINGULAR AND STRICTLY COSINGULAR OPERATORS BY PERTURBATION CLASSES
2011
AbstractWe consider a class of operators that contains the strictly singular operators and it is contained in the perturbation class of the upper semi-Fredholm operators PΦ+. We show that this class is strictly contained in PΦ+, solving a question of Friedman. We obtain similar results for the strictly cosingular operators and the perturbation class of the lower semi-Fredholm operators PΦ−. We also characterize in terms of PΦ+ and in terms of PΦ−. As a consequence, we show that and are the biggest operator ideals contained in PΦ+ and PΦ−, respectively.
Weyl type theorems for bounded linear operators on Banach spaces
2011
In 1909 H. Weyl [59] studied the spectra of all compact linear perturbations of a self-adjoint operator defined on a Hilbert space and found that their intersection consisted precisely of those points of the spectrum where are not isolated eigenvalues of nite multiplicity. Later, the property established by Weyl for self-adjoint operators has been observed for several other classes of operators, for instance hyponormal operators on Hilbert spaces, Toeplitz operators,convolution operators on group algebras, and many other classes of operators ned on Banach spaces . In the literature, a bounded linear operator defined on a Banach space which satisfies this property is said to satisfy Weyl's t…
Property (w) and perturbations III
2009
AbstractThe property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core K(λ0I−T)={0} for some λ0∈C. The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.
Weyl-Type Theorems
2018
In the previous chapters we introduced several classes of operators which have their origin in Fredholm theory. We also know that the spectrum of a bounded linear operator T on a Banach space X can be split into subsets in many different ways.
Some Remarks on the Spectral Properties of Toeplitz Operators
2019
In this paper, we study some local spectral properties of Toeplitz operators $$T_\phi $$ defined on Hardy spaces, as the localized single-valued extension property and the property of being hereditarily polaroid.
On the spectral properties of some classes of operators
2010
This article concerns the spectral properties of some classes of operators defined by means some inequelities
Polaroid operators and Weyl type theorems
2015
Weyl type theorems have been proved for a considerably large number of classes of operators. In this work, after introducing the class of polaroid operators and some notions from local spectral theory, we determine a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. The theory is exemplified by given several examples of hereditarily polaroid operators.
Property (gR) and perturbations
2012
Property (gR) holds for a bounded linear operator T defined on a complex Banach space X, the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points c of the approximate point spectrum such cI -T is upper semi B-Browder. In this paper we consider the permanence of this property under nilpotent, perturbations commuting with T.
Single valued extension Property and Fredholm theory
2006
Examples of improjective operators
2000
It has been an open question for some time whether improjective operators are always inessential. Here we give some examples that answer in the negative this question as well as some other related ones, posed in [2, 3, 11, 12]. The description of the examples uses a indecomposable space, constructed by Gowers and Maurey [5], and a characterization of the indecomposable Banach spaces in terms of improjective operators.
Weyl-Type Theorems on Banach Spaces Under Compact Perturbations
2018
In this paper, we study Browder-type and Weyl-type theorems for operators $$T+K$$ defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.
Polaroid-Type Operators
2018
In this chapter we introduce the classes of polaroid-type operators, i.e., those operators T ∈ L(X) for which the isolated points of the spectrum σ(T) are poles of the resolvent, or the isolated points of the approximate point spectrum σap(T) are left poles of the resolvent. We also consider the class of all hereditarily polaroid operators, i.e., those operators T ∈ L(X) for which all the restrictions to closed invariant subspaces are polaroid. The class of polaroid operators, as well as the class of hereditarily polaroid operators, is very large. We shall see that every generalized scalar operator is hereditarily polaroid, and this implies that many classes of operators acting on Hilbert s…
SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS
1995
Abstract We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.
Weyl Type Theorems for Left and Right Polaroid Operators
2010
A bounded operator defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a unified way many recent results, concerning Weyl type theorems (generalized or not) for important classes of operators.
Local spectral theory for Drazin invertible operators
2016
Abstract In this paper we investigate the transmission of some local spectral properties from a bounded linear operator R, as SVEP, Dunford property (C), and property (β), to its Drazin inverse S, when this does exist.
Weil's theorem for perturbations of paranormal operators
2007
A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl''s theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl''s theorem for every algebraic operator K which commutes with T.
Operators which have a closed quasi-nilpotent part
2002
We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property.
On generalized a-Browder's theorem
2007
We characterize the bounded linear operators T satisfying generalized a-Browder's theorem, or generalized a-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H0(�I T) asbelongs to certain sets of C. In the last part we give a general framework in which generalized a-Weyl's theorem follows for several classes of operators. 1. Preliminaries. Let L(X) denote the space of bounded linear oper- ators on an infinite-dimensional complex Banach space X. For T ∈ L(X), denote by α(T) the dimension of the kernel ker T, and by β(T) the codi- mension of the range T(X). The operator T ∈ L(X) is called upper semi- Fredholm if α(T) < ∞ and T(X) is closed, and lower …
Projections and isolated points of parts of the spectrum
2018
In this paper, we relate the existence of certain projections, commuting with a bounded linear operator $T\in L(X)$ acting on Banach space $X$, with the generalized Kato decomposition of $T$. We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$. Further results are given for the isolated points of some parts of the spectrum.
Weyl's type theorems and perturbations
2007
SVEP and local spectral radius formula for unbounded operators
2014
In this paper we study the localized single valued extension property for an unbounded operator T. Moreover, we provide sufficient conditions for which the formula of the local spectral radius holds for these operators.
Weyl's theorems through local spectral theory
2004
Intrinsic characterizations of perturbation classes on some Banach spaces
2010
We investigate relationships between inessential operators and improjective operators acting between Banach spaces X and Y, emphasizing the case in which one of the spaces is a C(K) space. We show that they coincide in many cases, but they are different in the case X=Y =C(K 0), where K 0 is a compact space constructed by Koszmider. Mathematics Subject Classification (2000)47A53 KeywordsInessential operators-Improjective operators-Fredholm theory
Perturbations of polaroid type operators on Banach spaces and Applications
2011
We study the permanence of polaroid type conditions under perturbations
Property (w) for perturbations of polaroid operators
2008
Abstract A bounded linear operator T ∈ L ( X ) acting on a Banach space satisfies property ( w ) , a variant of Weyl’s theorem, if the complement in the approximate point spectrum σ a ( T ) of the Weyl essential approximate-point spectrum σ wa ( T ) is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property ( w ) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T.
On the operators which are invertible modulo an operator ideal
2001
Atkinson [3] studied the operators which are left invertible $i(X, Y) or right invertible $T{X, Y) modulo /C, with K. the compact operators. He proved that an operator T € C(X, Y) belongs to <£/ or $ r if and only if the kernel and the range of T are complemented and additionally, the kernel is finite dimensional or the range is finite codimensional, respectively. Yood [19] obtained some perturbation results for these classes and Lebow and Schechter [12] proved that the inessential operators form the perturbation class for $,(A") and $r{X). Yang [18] extended some results of ^3, 19] to operators invertible modulo W, with W the weakly compact operators. His aim was to study a generalised Fre…
Property (gb) through local spectral theory
2014
Property (gb) for a bounded linear operator T on a Banach space X means that the points c of the approximate point spectrum for which c I-T is upper semi B-Weyl are exactly the poles of the resolvent. In this paper we shall give several characterizations of property (gb). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gb) holds for large classes of operators and prove the stability of property (gb) under some commuting perturbations.
Variations on Weyl's theorem
2006
AbstractIn this note we study the property (w), a variant of Weyl's theorem introduced by Rakočević, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T*) coincide whenever T* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).
Weyl's and Browder's theorems through the quasi-nilpotent part
2006
Weyl and Browder type theorems are characterized by means the quasi-nilpotent part
A variation on Weyl's theorem
2005
In this note we study the property , a variant of Weyl's theorem introduced by Rakočević, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property for T (respectively ) coincide whenever (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property .
On Drazin invertibility
2008
The left Drazin spectrum and the Drazin spectrum coincide with the upper semi-B-Browder spectrum and the B-Browder spectrum, respectively. We also prove that some spectra coincide whenever T or T* satisfies the single-valued extension property.
Weyl's theoren, a-Weyl's theorem and single valued extension property
2005
Weyl's theorem for perturbations of paranormal operators
2007
A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl's theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T.
Generalized Weyl's theorem and quasi-affiniy.
2010
A bounded operator T in 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. In this paper we prove that generalized Weyl's theorem holds for several classes of operators, extending previous results obtained in [24] and [15]. We also consider the preservation of generalized Weyl's theorem between two operators T in L(X), S in L(Y ) in the case that these are intertwined by a quasi-affinity A in L(X; Y ), or in the more general case that T and S are asymptotically intertwined by A.
Incomparable Banach spaces and operator semigroups
2002
Using the notions of total incomparability and total coincomparability of Banach spaces, we define two families of operator semigroups. We show that these semigroups are minimal, in the sense that they admit a perturbative characterization. Moreover, they allow us to characterize the corresponding incomparability classes.
Quasi-Fredholm operators and localized SVEP
2007
Generalized Browder’s Theorem and SVEP
2007
A bounded operator \(T \in L(X), X\) a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H0(λI − T) as λ belongs to certain …
Property (w) and perturbations
2007
A bounded linear operator T ∈ L(X) defined on a Banach space X satisfies property (w), a variant of Weyl’s theorem, if the complement in the approximate point spectrum σa(T ) of the Weyl essential approximate spectrum σwa(T ) coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property (w), for a bounded operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operator and quasi-nilpotent operators commuting with T .
Single valued extension property and semi-Browder spectra
2004
On riesz and inessential operators
1989
A characterization of riesz operators
1987
Property (w) and perturbations II
2008
AbstractThis note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) 683–692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T. A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T, also under the assumption that T is a-isoloid.
A Unifying Approach to Weyl Type Theorems for Banach Space Operators
2013
Weyl type theorems have been proved for a considerably large number of classes of operators. In this paper, by introducing the class of quasi totally hereditarily normaloid operators, we obtain a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. This framework also entails Weyl type theorems for perturbations f(T + K), where K is algebraic and commutes with T, and f is an analytic function, defined on an open neighborhood of the spectrum of T + K, such that f is non constant on each of the components of its domain.
Browder's theorems through localized SVEP
2005
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.
The perturbation classes problem for closed operators
2017
We compare the perturbation classes for closed semi-Fredholm and Fredholm operators with dense domain acting between Banach spaces with the corresponding perturbation classes for bounded semi-Fredholm and Fredholm operators. We show that they coincide in some cases, but they are different in general. We describe several relevant examples and point out some open problems.
Property (w) under compact or Riesz perturbations
2010
Si studia la permanenza della proprietà (w), una variante del Teorema di Weyl, nel caso che un operatore sia perturbato da un operatore compatto oppure di Riesz