Single-valued Extension Property Research Articles

Common local spectral properties for the operator products

In their paper [Generalized Clines Formula for g-Drazin Inverse in a Ring, Filomat 35(8) (2021) 2573–2583], Chen and Abdolyousefi examined the common spectral properties of operators [Formula: see text] and [Formula: see text], subject to the following conditions: [Formula: see text] [Formula: see text] In this note, we explore the common local spectral properties of these operators. We demonstrate that [Formula: see text] and [Formula: see text] shared the single-valued extension property, the Bishop property [Formula: see text], the property [Formula: see text], the decomposition property [Formula: see text], and decomposability. Furthermore, we investigate the closedness of the analytic core and the quasinilpotent part. We also delve into the Dunford’s property [Formula: see text] and the property [Formula: see text].

Weyl’s type theorems for linear relations satisfying the single valued extension property

Weyl’s type theorems for linear relations satisfying the single valued extension property

Square roots of complex symmetric operators

In this paper we study square roots of complex symmetric operators. In particular, we prove that if T∈L(H) is a square root of a complex symmetric operator, then T∗ has the single-valued extension property if and only if so does T. Moreover, in this case, T has the Bishop's property (β) if and only if T is decomposable. Finally, we show that if T has a nontrivial hyperinvariant subspace, then T∗ has a nontrivial invariant subspace.

Local Ergodic Theorems for $$C_0$$-Semigroups

Let \(\{T(t)\}_{t\ge 0}\) be a \(C_0\)-semigroup of bounded linear operators on the Banach space X into itself and let A be their infinitesimal generator. In this paper, we show that if T(t) is uniformly ergodic, then A does not have the single valued extension property, which implies that A must have a non-empty interior of the point spectrum. Furthermore, we introduce the local mean ergodic for \(C_0\)-semigroup T(t) at a vector \(x\in X\) and we establish some conditions implying that T(t) is a local mean ergodic at x.

Necessary conditions for extended spectral decomposable multivalued linear operators

In this paper, we use subsets of the Riemann sphere and specific types of invariant linear subspaces to introduce the extended spectral decomposable multivalued linear operators (linear relations) in Banach spaces. We also introduce the extended Bishop's property, the extended relatively single-valued extension property and the extended Dunford's property. More importantly, we show that these properties are three necessary conditions for a linear relation to be extended spectral decomposable.

On local spectral properties of operator matrices

In this paper, we focus on a 2 times 2 operator matrix T_{epsilon _{k}} as follows: \t\t\tTϵk=(ACϵkDB),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} T_{\\epsilon _{k}}= \\begin{pmatrix} A & C \\\\ \\epsilon _{k} D & B\\end{pmatrix}, \\end{aligned}$$ \\end{document} where epsilon _{k} is a positive sequence such that lim_{krightarrow infty }epsilon _{k}=0. We first explore how T_{epsilon _{k}} has several local spectral properties such as the single-valued extension property, the property (beta ), and decomposable. We next study the relationship between some spectra of T_{epsilon _{k}} and spectra of its diagonal entries, and find some hypotheses by which T_{epsilon _{k}} satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix T_{epsilon _{k}} has a nontrivial hyperinvariant subspace.

Property (ω) and the Single-valued Extension Property

Property (ω) and the Single-valued Extension Property

The Local Spectral Theory for Linear Relations Involving SVEP

Motivated by Aiena, Trapani and Triolo’s work (Filomat 28(2), 263–273, 2014), we introduced some concepts of the local spectral theory and the single-valued extension property abbreviated SVEP of the closed linear relations on a Banach space. After that, we analyzed basic proprieties of these notions and established a relationship between the analytic spectral subspace and the analytic core (see Theorem 3.5).

Back to the common spectral properties of operators satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

In [A. Gupta and A. Kumar, Common spectral properties of linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text], Asian-European J. Math. 12(5) (2019) 1950084], Gupta and Kumar studied the common spectral properties for two Banach bounded linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text] for some positive integer [Formula: see text]. In this note, we study the common local spectral properties for such operators by showing that [Formula: see text] and [Formula: see text] share the single valued extension property, the Bishop property [Formula: see text] the property [Formula: see text] the decomposition property [Formula: see text] and decomposability. The closedness of analytic core and the quasinilpotent part and then the Dunford’s property [Formula: see text] and the property [Formula: see text] are also investigated.

Left and right generalized Drazin inverses for closed operators and application to singular linear equations

We define and study left and right generalized Drazin inverses for a closed densely defined linear operator in a Hilbert space. These operators are characterized by means of the generalized Kato decomposition and the single-valued extension property. We prove that they are invariant under commuting quasinilpotent perturbations. Finally, we give an application to solve singular linear equations.

On the Single-Valued Extension Property of Hyponormal Operators on Banach Spaces

In this paper, we study the single-valued extension property of hyponormal operators on Banach spaces $${\mathcal {X}}$$ . In particular, we prove that if a bounded linear operator T on $${\mathcal {X}}$$ has the property (II) or the property (I $$')$$ (see Definition 2.3), then T has the single-valued extension property. Moreover, we show that for strictly convex (resp., smooth) $$\mathcal {{\mathcal {X}}}$$ , if $$T\in {{\mathcal {L}}}(\mathcal {{\mathcal {X}}})$$ is hyponormal (resp., $$\,^*$$ -hyponormal) on $$\mathcal {{\mathcal {X}}}$$ , then T has the single-valued extension property.

M-hypercyclicity of C 0-semigroup and Svep of its generator

Abstract Let 𝒯 = (Tt ) t ≥0 be a C 0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C 0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C 0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C 0-semigroup to be M-hypercyclic.

On generalized Drazin-Riesz operators

In this paper, we investigate the classes of operators as class of generalized Drazin Riesz operators. We give some results for these classes throught localized single valued extension property (SVEP). Some applications are given.

Perturbation of the spectra of complex symmetric operators

An operator T on a complex Hilbert space H is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for H. This paper focuses on the perturbation theory for the spectra of complex symmetric operators. We prove that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and having the single-valued extension property. Also it is proved that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and satisfying generalized Weyl?s theorem.

Property $(w)$ of upper triangular operator Matrices

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

Further common local spectral properties for bounded linear operators

We study common local spectral properties for bounded linear operators A ∈ ℒ(X,Y) and B,C ∈ ℒ (Y,X) such that A(BA)2=ABACA=ACABA=(AC)2A. We prove that AC and BA share the single valued extension property, the Bishop property (β), the property (β e), the decomposition property (δ) and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.

Generalized Kato-meromorphic decomposition, generalized Drazin-meromorphic invertible operators and single-valued extension property

In this paper we define and study generalized Kato-meromorphic decomposition and generalized Drazin-meromorphic invertible operators. A bounded linear operator T on a Banach space X is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator S acting on X such that $$TS=ST$$ , $$STS=S$$ , $$ TST-T$$ is meromorphic. Among others, we show that T is generalized Drazin-meromorphic invertible if and only if T admits a generalized Kato-meromorphic decomposition and 0 is not an interior point of $$\sigma (T)$$ , and this is also equivalent to the fact that T is a direct sum of a meromorphic operator and an invertible operator. Also we study bounded linear operators which can be expressed as a direct sum of a meromorphic 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 $$\lambda _0\in \mathbb {C}$$ in the case that $$\lambda _0-T$$ admits a generalized Kato-meromorphic decomposition, and as a consequence we get several results on cluster points of some distinguished part of the spectrum. Furthermore, we investigate corresponding spectra and prove that these spectra are empty if and only if the operator T is polynomially meromorphic.

Left and right generalized Drazin invertible operators and local spectral theory

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse.

Some Remarks on the Spectral Properties of Toeplitz Operators

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.

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.