Single-valued Extension Property Research Articles

Local spectral property of 2 x 2 operator matrices

In this paper we study the local spectral properties of 2 x 2 operator matrices. In particular, we show that every 2 x 2 operator matrix with three scalar entries has the single valued extension property. Moreover, we consider the spectral properties of such operator matrices. Finally, we show that some of such operator matrices are decomposable.

Some properties of (m,C)-isometric operators

In this paper, we study if T is an (m,C)-isometric operator and CT+C commutes with T, then T+ is an (m,C)-isometric operator. We also give local spectral properties and spectral relations of (m;C)-isometric operators, such as property (?), decomposability, the single-valued extension property and Dunford?s boundedness. We also investigate perturbation of (m,C)-isometric operators by nilpotent operators and by algebraic operators and give some properties.

Generalized Weyl\u2019s theorem and property (gw) for upper triangular operator matrices

It is known that if Ain mathscr {L}(mathscr {X}) and Bin mathscr {L}(mathscr {Y}) are Banach operators with the single-valued extension property, SVEP, then the matrix operator M_mathrm{{C}}=begin{pmatrix} A &{} C 0&{} B end{pmatrix} has SVEP for every operator Cin mathscr {L}(mathscr {Y},mathscr {X}), and hence obeys generalized Browder’s theorem. This paper considers conditions on operators A, B, and M_0 ensuring generalized Weyl’s theorem and property (Bw) for operators M_mathrm{{C}}. Moreover, certain conditions are explored on Banach space operators T and S so that Toplus S obeys property (gw).

Compact perturbations resulting in hereditarily polaroid operators

A Banach space operator $A\in B({\cal X})$ is polaroid, $A\in(\cal P)$, if the isolated points of the spectrum $\sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in(\cal {HP})$, if every restriction of $A$ to a closed invariant subspace is polaroid. It is seen that operators $A\in(\cal {HP})$ have SVEP - the single-valued extension property - on $\Phi_{sf}(A)=\{\lambda: A-\lambda$ is semi Fredholm $\}$. Hence $\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A), \ind(A-\lambda)>0\}=\emptyset$ for operators $A\in(\cal {HP})$, and a necessary and sufficient condition for the perturbation $A+K$ of an operator $A\in B({\cal X})$ by a compact operator $K\inB({\cal X})$ to be hereditarily polaroid is that $\Phi_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal X})$ to have SVEP on $\Phi_{sf}(A)$ is that its component $\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \ind(A-\lambda)\leq 0\}$ is connected. We prove: If $A\in B({\cal H})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K\in B({\cal H})$ such that $A+K\in(\cal {HP})$ is that $\Omega_a(A)$ is connected.

Bounded Local Resolvents for Operators with Single-Valued Extension Property

Let T be a bounded operator with (SVEP) on its localizable spectrum \(\sigma _\mathrm{loc}(T)\). We show that for every open subset U of \(\sigma _\mathrm{loc}(T)\), there exists a unit vector x whose local spectrum coincides with the closure of U, and such that its local resolvent function is bounded. This result answers positively to an open question stated by several authors, and extends the both cases of operators with trivial divisible subspace and operators whose point spectrum has empty interior.

Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators

We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kato–Riesz decomposition if there exists a pair of T-invariant closed subspaces (M, N) such that , the reduction is Kato and is Riesz. In this paper, we define and investigate the generalized Kato–Riesz spectrum of an operator. For T is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator S acting on X such that , , is Riesz. We investigate generalized Drazin-Riesz invertible operators and also characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular, we characterize the single-valued extension property at a point in the case that admits a generalized Kato–Riesz decomposition.

Subscalarity, Invariant, and Hyperinvariant Subspaces for Upper Triangular Operator Matrices

In this paper, we introduce the class of quasi-nM-hyponormal operators and study various properties. We show that every quasi-nM-hyponormal operator is subscalar of order 2n; in particular, every M-hyponormal operator is subscalar of order two. Consequently, if the spectrum of T has a nonempty interior in \(\mathbb {C}\), then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for quasi-nM-hyponormal operators. Finally, we consider some of the spectral properties of this class and show that they share many spectral properties with normal operators on Hilbert spaces, mainly concerning Fredholm theory and local spectral theory. We are primarily interested in isolated points of the spectra of quasi-nM-hyponormal operators as well as the isolated points of the approximate point spectra. These properties lead to the concept of a polaroid-type operator, which when combined with the single-valued extension property, an important property in local spectral theory, produces a general framework for several versions of Weyl-type theorems.

On (m, C)-Isometric Operators

In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if $$T\in \mathcal{{L(H)}}$$ is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if $$T^{*}$$ has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.

On the property $$(Z_{E_{a}})$$ ( Z E a )

The paper introduces the notion of properties \((Z_{\Pi _{a}})\) and \((Z_{E_{a}})\) as variants of Weyl’s theorem and Browder’s theorem for bounded linear operators acting on infinite dimensional Banach spaces. A characterization of these properties in terms of localized single valued extension property is given, and the perturbation by commuting Riesz operators is also studied. Classes of operators are considered as illustrating examples.

Weak supercyclicity: dynamics of paranormal operators

If A is a Banach space operator with the single-valued extension property such that the isolated points of the spectrum of A are poles of (the resolvent of) A, then A is neither weakly hypercyclic nor n-supercyclic for any positive integer \(n\ge 1\). Furthermore, if also A and \(A^{-1}\) (whenever it exists) are normaloid, then A weakly supercyclic implies A is an invertible isometry. Translated to classes of Hilbert space operators, this implies that if an operator A is either paranormal or \(*\)-paranormal (or even M-hyponormal), then it is not weakly hypercyclic or n-supercyclic; furthermore, A weakly supercyclic implies A is a scalar multiple of a unitary operator [thereby answering a problem posed in Duggal et al. (J Math Anal Appl 427:107–113, 2015)].

Perturbation of single-valued extension property and its application

Perturbation of single-valued extension property and its application

The localized single-valued extension property and Riesz operators

The localized single-valued extension property is stable under commuting Riesz perturbations.

Complex symmetric weighted composition operators on [formula omitted

In this paper we study complex symmetric weighted composition operators on the Hardy space. We provide some characterizations of ψ and φ when a weighted composition operator Wψ,φ is complex symmetric. We investigate which combinations of weights ψ and maps of the open unit disk φ give rise to complex symmetric weighted composition operators with a special conjugation. As some applications, we obtain several examples for nonnormal complex symmetric operators. In addition, we give spectral properties of complex symmetric weighted composition operators. We examine eigenvalues and eigenvectors of such operators and find some conditions for which a complex symmetric weighted composition operator is Hilbert–Schmidt. Finally, we consider cyclicity, hypercyclicity, and the single-valued extension property for complex symmetric weighted composition operators.

Composition operators for which and commute

We define . In this paper, we characterize composition operators and their adjoints which belong to , where the maps are linear fractional selfmaps of the open unit disk into itself. If is an automorphism of or , then the case for is precisely when it is normal. When , we also prove that if , then either or , which implies that the only binormal composition operators with and are normal. Moreover, we show that if and is not normal, then implies that and is neither real nor purely imaginary, while ensures that and is real. Finally, we study composition operators in where is an analytic selfmap into . In particular, this operator has the single-valued extension property.

On generalized Weyl’s type theorem

It is shown that if a bounded linear operator T or its adjoint T* has the single-valued extension property, then generalized Browder’s theorem holds for f(T) for every f ∈ H(σ(T)). We establish the spectral theorem for the B-Weyl spectrum which generalizes [15, Theorem 2.1] and we give necessary and sufficient conditions for such operator T to obey generalized Weyl’s theorem.

SVEP and local spectral radius formula for unbounded operators

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.

Regularities, Single Valued Extension Property and Spectral Mapping Theorem

The present paper reviews spectral mapping theorems for 32 nonempty spectra by using the theory of regularities and semiregularities and the theory of single valued extension property.

Properties (t) and (gt) for Bounded Linear Operators

In this paper we introduce and study the properties (t) and (gt), which extend properties (w) and (gw). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (t) and property (gt) hold. We also relate these properties with Weyl’s type theorems. We show that if T is a bounded linear operator acting on a Banach space \({\fancyscript{X}}\), then property (gt) holds for T if and only if property (gw) holds for T and σ(T) = σa(T). Analogously, we show that property (t) holds for T if and only if property (ω) holds for T and σ(T) = σa(T). We also study the properties (t) and (gt) for the operators satisfying the single valued extension property. Moreover, these properties are also studied in the framework of polaroid operators.

Local spectral radius formulas for a class of unbounded operators on Banach spaces

We exhibit unbounded operators on Banach spaces having the single-valued extension property, for which the local spectrum at suitable points can be determined, and for which a local spectral radius formula holds, analogous to that for a bounded operator on a Banach space with the singlevalued extension property. Such an operator can occur as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform which is an isomorphism of topological vector spaces between the domain, in its own topology, and a sequence space. We give examples (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.

Property (aw) and Weyl’s Theorem

In this note we study the property (aw), a variant of Weyl’s theorem introduced by Berkani and Zariouh, 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 (aw) holds. We also relate this property with Weyl’s theorem, a-Weyl’s theorem and property (w). Finally, we show that if T is a-polaroid and either T or T* has SVEP then f(T) satisfies property (aw) for each \({f \in H_1(\sigma(T))}\) .