Single-valued Extension Property Research Articles

On the stability of the localized single-valued extension property under commuting perturbations

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.

Taylor exactness, SVEP and spectral mapping theorems

� 2013 Muneo Cho and Robin Harte. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Spectral mapping theorems for local spectra derived from the holomorphic range and the single valued extension property are proved with the aid of Taylor exactness.

Propertiesandfor Bounded Linear Operators

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

Local Spectrum of a Family of Operators

‎Starting from the classic definitions of resolvent set and spectrum‎ ‎of a linear bounded operator on a Banach space‎, ‎we introduce the‎ ‎local resolvent set and local spectrum‎, ‎the local spectral space and‎ ‎the single-valued extension property of a family of linear bounded‎ ‎operators on a Banach space‎. ‎Keeping the analogy with the classic‎ ‎case‎, ‎we extend some of the known results from the case of a linear‎ ‎bounded operator to the case of a family of linear bounded operators‎ ‎on a Banach space.

Components of generalized Kato resolvent set and single-valued extension property

In this paper, we use the constancy of certain subspace valued mappings on the components of the generalized Kato resolvent set and the equivalences of the single-valued extension property at a point 0 for operators which admit a generalized Kato decomposition to obtain a classification of the components of the generalized Kato resolvent set of operators. We also give some applications of these results.

Topological uniform descent and localized SVEP

A localized version of the single-valued extension property is studied, for a bounded linear operator T acting on a Banach space and its adjoint T⁎, at the points λ0∈C such that λ0I−T has topological uniform descent (TUD for brevity). We characterize the single-valued extension property at these points for T and T⁎. We also give some applications of these results. As we give a counterexample to show that the adjoint of an operator with TUD is not necessarily with TUD, it is worth to mention that the characterizations of SVEP at these points for T⁎ cannot be obtained dually from the characterizations of SVEP at the same points for T. It is quite different from the case that λ0I−T is of Kato type or quasi-Fredholm.

The stability of the single valued extension property

A Hilbert space operator T has the single valued extension property if the only analytic function f which satisfies (T−λI)f(λ)=0 is f≡0. Clearly the point spectrum of any operator which has empty interior must have the single valued extension property. Using the induced spectrum of “consistent in Fredholm and index”, we investigate the stability of single valued extension property under compact perturbations, and we characterize those operators for which the single valued extension property is stable under compact perturbations.

On (A,m)-expansive operators

We give several conditions for $(A,m)$-expansive operators to have the single-valued extension property. We also provide some spectral properties of such operators. Moreover, we prove that the $A$-covariance of any $(A,2)$-expansive operator $T\in\mathcal

The stability of the single-valued extension property

A Hilbert space operator T ∈ B ( H ) may be said to be “consistent in Fredholm and index” provided that for each S ∈ B ( H ) , one of the following cases occurs: (1) T S and S T are Fredholm together and ind ( T S ) = ind ( S T ) = ind ( S ) ; (2) both T S and S T are not Fredholm. The induced spectrum contributes the conditions for the stability of the single-valued extension property under compact perturbations. We characterize those operators for which the single-valued extension property is stable under compact perturbations.

Spectral properties of commuting operations for 𝑛-tuples

LetR\textbf {R}andS\mathbf { S}be commutingnn-tuples. We give some spectral and local spectral relations betweenRS\mathbf { RS}andSR\mathbf { SR}. In particular, we show thatRS\mathbf { RS}has the single valued extension property or satisfies Bishop’s property(β)(\beta )if and only ifSR\mathbf { SR}has the corresponding property.

SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T ⁎ , the adjoint of T, is quasitriangular. Moreover, if T ⁎ is quasitriangular, then, given an ε > 0 , there exists a compact operator K on H with ‖ K ‖ < ε such that T + K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.

B-Fredholm and Drazin invertible operators through localized SVEP

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 $\lambda \in \mathbb C$ such that $T$ does not have the single-valued extension property at $\lambda $. 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.

UPPER TRIANGULAR OPERATORS WITH SVEP

A Banach space operator A <TEX>$\in$</TEX> B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A <TEX>$\in$</TEX> (<TEX>$\mathcal{H}\mathcal{P}$</TEX>), if every part of A is polaroid. Let <TEX>$X^n\;=\;\oplus^n_{t=i}X_i$</TEX>, where <TEX>$X_i$</TEX> are Banach spaces, and let A denote the class of upper triangular operators A = <TEX>$(A_{ij})_{1{\leq}i,j{\leq}n$</TEX>, <TEX>$A_{ij}\;{\in}\;B(X_j,X_i)$</TEX> and <TEX>$A_{ij}$</TEX> = 0 for i > j. We prove that operators A <TEX>$\in$</TEX> A such that <TEX>$A_{ii}$</TEX> for all <TEX>$1{\leq}i{\leq}n$</TEX>, and <TEX>$A^*$</TEX> have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A <TEX>$\in$</TEX> A such that <TEX>$A_{ii}$</TEX> <TEX>$\in$</TEX> (<TEX>$\mathcal{H}\mathcal{P}$</TEX>) for all <TEX>$1{\leq}i{\leq}n$</TEX> are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that <TEX>$\oplus^n_{i=1}R_{ii}$</TEX> is a Riesz operator, which commutes with A.

Browder and Weyl spectra of upper triangular operator matrices

Let MC = (A/0 C/B) ( B(X ( X ) be an upper triangulat Banach space operator. The relationship between the spectra of MC and M0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators MC and M0 satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type 'M0 satisfies a-Browder's (or a-Weyl's) theorem ( MC satisfies a-Browder's (resp., a-Weyl's) theorem' to hold. 2010 Mathematics Subject Classifications. Primary 47B47, 47A10, 47A11. .

Polaroid Operators with SVEP and Perturbations of Property (gw)

Let \({\mathcal{L}(X)}\) be the algebra of all bounded linear operators on X and \({\mathcal{P}S(X)}\) be the class of polaroid operators with the single-valued extension property. The property (gw) holds for \({T \in \mathcal{L}(X)}\) if the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues of the spectrum. In this note we focus on the stability of the property (gw) under perturbations: we prove that, if \({T \in \mathcal{P}S(X)}\) and A (resp. Q) is an algebraic (resp. quasinilpotent) operator, then the property (gw) holds for f(T* + A*) (resp. f(T* + Q*)) for every analytic function f in σ(T + A) (resp. σ(T + Q)). Some applications are also given.

Essential Spectral Inclusions for Operators on Banach Spaces

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context.

Generalized Kato decomposition, single-valued extension property and approximate point spectrum

In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property 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 0 ∈ C in the case that λ 0 I − T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.

A Note on the Browder’s and Weyl’s theorem

Let T be a Banach space operator, E(T) be the set of all isolated eigenvalues of T and π(T) be the set of all poles of T. In this work, we show that Browder’s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl’s theorem for operator satisfying E(T) = π(T). An application is also given.

On the Property (gw)

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakocevic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π a (T) = E(T), where Π a (T) is the set of left poles of T and E(T) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples.

Upper triangular operator matrices with the single-valued extension property

If A = ( A i j ) 1 ⩽ i , j ⩽ n ∈ B ( X ) is an upper triangular Banach space operator such that A i i A i j = A i j A j j for all 1 ⩽ i ⩽ j ⩽ n , then A has SVEP or satisfies (Dunford's) condition ( C ) or (Bishop's) property ( β ) or (the decomposition) property ( δ ) if and only if A i i , 1 ⩽ i ⩽ n , has the corresponding property.