scholarly journals Results of Semigroup of Linear Operator in Spectral Theory

2021 ◽  
Keyword(s):  
Linear Operator ◽  
Spectral Theory

scholarly journals A characterisation and two examples of Riesz operators

1968 ◽  
Vol 9 (2) ◽  
pp. 106-110 ◽  
Author(s):  
T. A. Gillespie ◽  
T. T. West
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Spectral Projection ◽  
Resolvent Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

scholarly journals The minimum modulus of a linear operator and its use in spectral theory

1962 ◽  
Vol 22 (1) ◽  
pp. 15-41 ◽  
Author(s):  
H. Gindler ◽  
A. Taylor
Keyword(s):  
Linear Operator ◽  
Spectral Theory ◽  
Minimum Modulus

scholarly journals Some Properties of Spectral Theory in Fuzzy Hilbert Spaces

2017 ◽  
Vol 8 (2) ◽  
pp. 1-7
Author(s):  
Noori F. Al-Mayahi ◽  
Abbas M. Abbas
Keyword(s):  
Linear Operator ◽  
Spectral Theory ◽  
Hilbert Spaces ◽  
Normed Spaces

In this paper we give some definitions and properties of spectral theory in fuzzy Hilbert spaces also we introduce  definitions Invariant  under a linear operator  on fuzzy normed spaces and reduced  linear operator on fuzzy Hilbert spaces and we prove theorms  related to eigenvalue and eigenvectors ,eigenspace in fuzzy normed , Invariant and  reduced in fuzzy Hilbert spaces  and  show relationship between them.

scholarly journals Behavior of the Eigenvalues and Eignfunctions of the Regge-Type Problem

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 139
Author(s):  
Karwan H. F. Jwamer ◽  
Rando R. Q. Rasul
Keyword(s):  
Linear Operator ◽  
Spectral Theory ◽  
Infinite Number ◽  
Discontinuous Coefficients ◽  
Transmission Conditions ◽  
Type Problem

This article investigates the spectral theory of the problem of the Regge-type with transmission conditions and discontinuous coefficients. We formulate a new linear operator, by which we can deal with simplicity and boundedness of the eigenpairs of the problem. The aim of this work is to conduct that the problem has an infinite number of simple positive eigenvalues.

A Spectral Theory for Duality Systems of Operators on a Banach Space

1970 ◽  
Vol 22 (5) ◽  
pp. 994-996 ◽  
Author(s):  
J. G. Stampfli
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Banach Theorem ◽  
Duality Map ◽  
Image Position

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.A duality system for the operator T is an ordered sextuple(i) T is a bounded linear operator mapping the Banach space B into B,(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

Coupled operator systems and multiparameter spectral theory

1978 ◽  
Vol 80 (1-2) ◽  
pp. 23-34 ◽  
Author(s):  
G. F. Roach ◽  
B. D. Sleeman
Keyword(s):  
Linear Operator ◽  
Spectral Theory ◽  
Column Vector ◽  
Parseval Equality ◽  
Complex Scalar ◽  
Expansion Theorem ◽  
Operator Systems ◽  
Image Position

SynopsisIn this paper a spectral theory for completely coupled linear operator systems is developed. These systems take the formwhere Ak, Bk are n × n matrices with operator entries. Λ is an n × n matrix with complex scalar entries and xk is an n × 1 column vector. The main result is a Parseval equality and expansion theorem.

scholarly journals On the spectrum of a linear operator

1972 ◽  
Vol 13 (2) ◽  
pp. 98-101
Author(s):  
Michael B. Dollinger ◽  
Kirti K. Oberai
Keyword(s):  
Linear Operator ◽  
Compact Operator ◽  
Spectral Theory ◽  
Normed Spaces ◽  
Spectrum Of An Operator ◽  
Underlying Space ◽  
Definition Of ◽  
The Relationship

In the definition of the spectrum of a linear operator, it is customary to assume that the underlying space is complete. However there are occasions for which it is neither desirable nor necessary to assume completeness in order to obtain a spectral theory for an operator; for example, completeness is not needed in the Riesz theory of a compact operator (see e.g. [1: XI. 3]). Several non-equivalent definitions for the spectrum of an operator on normed spaces have appeared in the literature. We shall discuss the relationship among these definitions and some of the difficulties that arise in applying these definitions to obtain a spectral theory.

scholarly journals Property (gab) through localized SVEP

2015 ◽  
Vol 1 (2) ◽  
pp. 91-107 ◽  
Author(s):  
Pietro Aiena ◽  
Salvatore Triolo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Large Classes ◽  
Browder’S Theorem ◽  
Localized Svep ◽  
The Stability ◽  
Local Spectral Theory

AbstractIn 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.

scholarly journals Spectral theory for polynomially demicompact operators

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2017-2030
Author(s):  
Fatma Brahim ◽  
Aref Jeribi ◽  
Bilel Krichen
Keyword(s):  
Linear Operator ◽  
Spectral Theory ◽  
Fredholm Operators ◽  
Essential Spectra ◽  
Densely Defined

In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.

Semigroups of operators and an application to spectral theory

1984 ◽  
Vol 96 (1) ◽  
pp. 143-149 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Continuous Linear Operator ◽  
Integral Transforms ◽  
Fundamental Importance ◽  
Semigroups Of Operators ◽  
Continuous Linear ◽  
Scalar Type

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

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