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).

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.

On Certain Classes of Bounded Linear Operators

1970 ◽  
Vol 13 (4) ◽  
pp. 469-473
Author(s):  
C-S Lin
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

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 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 A note on the perturbation bounds of W-weighted Drazin inverse of linear operator in Banach space

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 505-511 ◽  
Author(s):  
Xue-Zhong Wang ◽  
Hai-Feng Ma ◽  
Marija Cvetkovic
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Perturbation Bound ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Perturbation Bounds ◽  
Weighted Drazin Inverse

We investigate the perturbation bound of the W-weighted Drazin inverse for bounded linear operators between Banach spaces and present two explicit expressions for the W-weighted Drazin inverse of bounded linear operators in Banach space, which extend the results in Chin. Anna. Math., 21C:1 (2000) 39-44 by Wei.

scholarly journals Note on the Invariance Properties of Operator Products Involving Generalized Inverses

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaoji Liu ◽  
Miao Zhang ◽  
Yaoming Yu
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Generalized Inverses ◽  
Operator Product ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Invariance Properties

We investigate further the invariance properties of the bounded linear operator productAC1 B1 Dand its range with respect to the choice of the generalized inversesXandYof bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.

Matrix mappings and general bounded linear operators on the space bv

2018 ◽  
Vol 68 (2) ◽  
pp. 405-414
Author(s):  
Ivana Djolović ◽  
Katarina Petković ◽  
Eberhard Malkowsky
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Hausdorff Measures ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Matrix ◽  
The Difference

Abstract If X and Y are FK spaces, then every infinite matrix A ∈ (X, Y) defines a bounded linear operator LA ∈ B(X, Y) where LA(x) = Ax for each x ∈ X. But the converse is not always true. Indeed, if L is a general bounded linear operator from X to Y, that is, L ∈ B(X, Y), we are interested in the representation of such an operator using some infinite matrices. In this paper we establish the representations of the general bounded linear operators from the space bv into the spaces ℓ∞, c and c0. We also prove some estimates for their Hausdorff measures of noncompactness. In this way we show the difference between general bounded linear operators between some sequence spaces and the matrix operators associated with matrix transformations.

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1010-1030 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Growth Conditions ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

scholarly journals A direct proof of a theorem of West on sequences of Riesz operators

1974 ◽  
Vol 15 (2) ◽  
pp. 93-94
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Direct Proof ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

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