scholarly journals Properties (S) and (gS) for bounded linear operators

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1641-1652 ◽  
Author(s):  
M.H.M. Rashid
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Linear Operators ◽  
Finite Multiplicity ◽  
Bounded Linear ◽  
Weyl Type Theorems ◽  
Weyl Theorem ◽  
Polaroid Operator ◽  
Isolated Eigenvalues

An operator T acting on a Banach space X obeys property (R) if ?0a(T) = E0(T), where ?0a(T) is the set of all left poles of T of finite rank and E0(T) is the set of all isolated eigenvalues of T of finite multiplicity. In this paper we introduce and study two new properties (S) and (gS) in connection with Weyl type theorems. Among other things, we prove that if T is a bounded linear operator acting on a Banach space, then T satisfies property (R) if and only if T satisfies property (S) and ?0(T) = ?0a(T), where ?0(T) is the set of poles of finite rank. Also we show if T satisfies Weyl theorem, then T satisfies property (S). Analogous results for property (gS) are given. Moreover, these properties are also studied in the frame of polaroid operator.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

2019 ◽  
Vol 99 (2) ◽  
pp. 274-283
Author(s):  
AMANOLLAH ASSADI ◽  
MOHAMAD ALI FARZANEH ◽  
HAJI MOHAMMAD MOHAMMADINEJAD
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subspace ◽  
Invariant Subspaces ◽  
Weak Limit ◽  
Linear Operators ◽  
Sufficient Condition ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Rank Operator

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

scholarly journals On the sublinear operators factoring throughLq

2004 ◽  
Vol 2004 (50) ◽  
pp. 2695-2704
Author(s):  
Lahcène Mezrag ◽  
Abdelmoumene Tiaiba
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Positive Constant ◽  
Bounded Linear Operator ◽  
Bounded Operator ◽  
Linear Operators ◽  
Sublinear Operator ◽  
Bounded Linear ◽  
Sublinear Operators

Let0<p≤q≤+∞. LetTbe a bounded sublinear operator from a Banach spaceXinto anLp(Ω,μ)and let∇Tbe the set of all linear operators≤T. In the present paper, we will show the following. LetCbe a positive constant. For alluin∇T,Cpq(u)≤C(i.e.,uadmits a factorization of the formX→u˜Lq(Ω,μ)→MguLq(Ω,μ), whereu˜is a bounded linear operator with‖u˜‖≤C,Mguis the bounded operator of multiplication byguwhich is inBLr+(Ω,μ)(1/p=1/q+1/r),u=Mgu∘u˜andCpq(u)is the constant ofq-convexity ofu) if and only ifTadmits the same factorization; This is under the supposition that{gu}u∈∇Tis latticially bounded. Without this condition this equivalence is not true in general.

scholarly journals Relatively compact-like perturbations, essential spectra and application

2004 ◽  
Vol 77 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Khalid Latrach ◽  
J. Martin Paoli
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Neutron Transport ◽  
Linear Operators ◽  
Essential Spectra ◽  
Graph Norm ◽  
Bounded Linear ◽  
Neutron Transport Equations ◽  
Densely Defined ◽  
Fredholm Perturbations

AbstractThe purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. IfAdenotes a closed densely defined linear operator on a Banach spaceX, our approach consists principally in considering the class ofA-closable operators which, regarded as operators in ℒ(XA,X) (whereXAdenotes the domain ofAequipped with the graph norm), are contained in the set ofA-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

scholarly journals Spectral approximation theorems for bounded linear operators

1973 ◽  
Vol 8 (2) ◽  
pp. 279-287 ◽  
Author(s):  
W.S. Lo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Eigenvalue Problem ◽  
Linear Operators ◽  
Spectral Approximation ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Approximation Theorems ◽  
Generalized Eigenvectors

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

scholarly journals Asymptotically quasi-compact products of bounded linear operators

1973 ◽  
Vol 16 (3) ◽  
pp. 286-289 ◽  
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Fredholm Determinant ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

scholarly journals Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means

2019 ◽  
Vol 17 (1) ◽  
pp. 1703-1715 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Finite Rank ◽  
Necessary Conditions ◽  
Operator Ideal ◽  
Linear Operators ◽  
Approximation Numbers ◽  
Bounded Linear ◽  
Weighted Means ◽  
Generalized Cesáro Sequence Space

Abstract Let E be a generalized Cesáro sequence space defined by weighted means and by using s-numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

scholarly journals Weyl type theorems for selfadjoint operators on Krein spaces

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6001-6016
Author(s):  
Il An ◽  
Jaeseong Heo
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Krein Space ◽  
Linear Operators ◽  
Fredholm Theory ◽  
Bounded Linear ◽  
Selfadjoint Operators ◽  
Krein Spaces ◽  
Weyl Type Theorems ◽  
Kreĭn Space

In this paper, we introduce a notion of the J-kernel of a bounded linear operator on a Krein space and study the J-Fredholm theory for Krein space operators. Using J-Fredholm theory, we discuss when (a-)J-Weyl?s theorem or (a-)J-Browder?s theorem holds for bounded linear operators on a Krein space instead of a Hilbert space.

On Meromorphic Operators, I

1967 ◽  
Vol 19 ◽  
pp. 723-736 ◽  
Author(s):  
S. R. Caradus
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Finite Multiplicity ◽  
Bounded Linear Operators ◽  
Zero Eigenvalue ◽  
Bounded Linear ◽  
Zero Points

If X is a complex Banach space and B(X) denotes the space of bounded linear operators on X, then the class of meromorphic operators consists of those T in B(X) such that the non-zero points of σ(T) are poles of the resolvent Rλ(T). If we also require that each non-zero eigenvalue of T have finite multiplicity, members of the class ⊆ so defined have been called operators of Riesz type. and have been studied in (2, 6, 7) and (1,4) respectively.

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