scholarly journals Single valued extension property and generalized Weyl’s theorem

2006 ◽  
Vol 131 (1) ◽  
pp. 29-38 ◽  
Author(s):  
M. Berkani ◽  
N. Castro ◽  
S. V. Djordjević
Keyword(s):  
Extension Property ◽  
Weyl’S Theorem ◽  
Single Valued Extension Property ◽  
Generalized Weyl’S Theorem

scholarly journals Propertiesandfor Bounded Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. H. M. Rashid
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Extension Property ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Single Valued Extension Property ◽  
Bounded Linear ◽  
Weyl Type Theorems

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.

scholarly journals Browder and Weyl spectra of upper triangular operator matrices

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 111-130 ◽  
Author(s):  
B.P. Duggal
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Weyl’S Theorem ◽  
Single Valued Extension Property ◽  
Browder’S Theorem ◽  
Subject Classifications ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Mathematics Subject Classifications ◽  
Necessary And Sufficient

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. .

scholarly journals On local spectral properties of operator matrices

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Il Ju An ◽  
Eungil Ko ◽  
Ji Eun Lee
Keyword(s):  
Spectral Properties ◽  
Extension Property ◽  
Positive Sequence ◽  
Weyl’S Theorem ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Hyperinvariant Subspace ◽  
The Relationship

AbstractIn this paper, we focus on a $2 \times 2$ 2 × 2 operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D & B\end{pmatrix}, \end{aligned}$$ T ϵ k = ( A C ϵ k D B ) , where $\epsilon _{k}$ ϵ k is a positive sequence such that $\lim_{k\rightarrow \infty }\epsilon _{k}=0$ lim k → ∞ ϵ k = 0 . We first explore how $T_{\epsilon _{k}}$ T ϵ k has several local spectral properties such as the single-valued extension property, the property $(\beta )$ ( β ) , and decomposable. We next study the relationship between some spectra of $T_{\epsilon _{k}}$ T ϵ k and spectra of its diagonal entries, and find some hypotheses by which $T_{\epsilon _{k}}$ T ϵ k satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix $T_{\epsilon _{k}}$ T ϵ k has a nontrivial hyperinvariant subspace.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

Property (ω) and the Single-valued Extension Property

2021 ◽  
Vol 37 (8) ◽  
pp. 1254-1266
Author(s):  
Lei Dai ◽  
Xiao Hong Cao ◽  
Qi Guo
Keyword(s):  
Extension Property ◽  
Single Valued Extension Property

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

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.

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