Perturbation theory for a linear operator

1967 ◽  
Vol 63 (1) ◽  
pp. 11-20 ◽  
Author(s):  
J. H. Webb
Keyword(s):  
Perturbation Theory ◽  
Linear Operator ◽  
Banach Spaces ◽  
State Diagram ◽  
Linear Operators ◽  
General Linear ◽  
Normed Spaces ◽  
Closed Operators

AbstractWe extend certain results of the theory of closed operators in Banach spaces to general linear operators in normed spaces. A ‘state diagram’ for linear operators is drawn up. We prove some perturbation theorems, improving or correcting certain results of Goldberg.

scholarly journals On Riesz operators

1969 ◽  
Vol 16 (3) ◽  
pp. 227-232 ◽  
Author(s):  
J. C. Alexander
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators ◽  
Image Position ◽  
Schauder’S Theorem

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

scholarly journals Compact diagonal linear operators on Banach spaces with unconditional bases

1993 ◽  
Vol 16 (4) ◽  
pp. 823-824
Author(s):  
Yun Sung Choi ◽  
Sung Guen Kim
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Uniform Continuity ◽  
Linear Operators ◽  
Unconditional Bases

LetEandFbe Banach spaces with equivalent normalized unconditional bases. In this note we show that a bounded diagonal linear operatorT:E→Fis compact if and only if its entries tend to0, using the concept of weak uniform continuity.

scholarly journals Quantities related to upper and lower semi-Fredholm type linear relations

2002 ◽  
Vol 66 (2) ◽  
pp. 275-289 ◽  
Author(s):  
Teresa Alvarez ◽  
Ronald Cross ◽  
Diane Wilcox
Keyword(s):  
Perturbation Theory ◽  
General Setting ◽  
Linear Operators ◽  
Normed Spaces ◽  
Fredholm Type ◽  
Linear Relations ◽  
Lower Semi ◽  
Strictly Singular

Certain norm related functions of linear operators are considered in the very general setting of linear relations in normed spaces. These are shown to be closely related to the theory of strictly singular, strictly cosingular, F+ and F− linear relations. Applications to perturbation theory follow.

scholarly journals The generalized condition numbers of bounded linear operators in Banach spaces

2004 ◽  
Vol 76 (2) ◽  
pp. 281-290 ◽  
Author(s):  
Guoliang Chen ◽  
Yimin Wei ◽  
Yifeng Xue
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Perturbation Analysis ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Condition Numbers ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Linear Operator Equations

AbstractFor any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

scholarly journals Polynomials on banach spaces whose duals are isomorphic to ℓ1 (Γ)

2004 ◽  
Vol 70 (1) ◽  
pp. 117-124 ◽  
Author(s):  
Raffaella Cilia ◽  
Maria D'Anna ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Fixed Integer ◽  
Integral Polynomial

We prove that the dual of a Banach space E is isomorphic to an ℓ1(Γ) space if and only if, for a fixed integer m, every m-homogeneous 1-dominated polynomial on E is nuclear. This extends a result for linear operators due to Lewis and Stegall. The same techniques used for this result allow us to prove that, if every m-homogeneous integral polynomial between two Banach spaces is nuclear, then every integral (linear) operator between the same spaces is nuclear.

scholarly journals Continuity and general perturbation of the Drazin inverse for closed linear operators

2002 ◽  
Vol 7 (6) ◽  
pp. 335-347 ◽  
Author(s):  
N. Castro González ◽  
J. J. Koliha ◽  
V. Rakocevic
Keyword(s):  
Asymptotic Behavior ◽  
Linear Operator ◽  
Error Estimates ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Closed Linear Operator ◽  
Operator Semigroups ◽  
Closed Operators ◽  
General Perturbation ◽  
Closed Linear Operators

We study perturbations and continuity of the Drazin inverse of a closed linear operatorAand obtain explicit error estimates in terms of the gap between closed operators and the gap between ranges and nullspaces of operators. The results are used to derive a theorem on the continuity of the Drazin inverse for closed operators and to describe the asymptotic behavior of operator semigroups.

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.

Regularizers of Closed Operators

1974 ◽  
Vol 17 (1) ◽  
pp. 67-71 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Banach Spaces ◽  
Compact Operator ◽  
Null Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Operators

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.

scholarly journals Perturbation theory of multivalued atkinson operators in normed spaces

2007 ◽  
Vol 76 (2) ◽  
pp. 195-204 ◽  
Author(s):  
Teresa Álvarez ◽  
Diane Wilcox
Keyword(s):  
Perturbation Theory ◽  
Linear Operators ◽  
Normed Spaces ◽  
Linear Relations ◽  
Small Norm ◽  
Additive Perturbation ◽  
Strictly Singular ◽  
Stability Results

We prove several stability results for Atkinson linear relations under additive perturbation by small norm, strictly singular and strictly cosingular multivalued linear operators satisfying some additional conditions.

scholarly journals Analytical Study on Approximate 𝝐-Birkhoff-James Orthogonality

2020 ◽  
pp. 1751-1758
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. Ahmed
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Analytical Study ◽  
Complete Characterization ◽  
Linear Operators ◽  
Minimum Norm ◽  
Bounded Linear ◽  
Rank One ◽  
James Orthogonality

In this paper, we obtain a complete characterization for the norm and the minimum norm attainment sets of bounded linear operators on a real Banach spaces at a vector in the unit sphere, using approximate 𝜖-Birkhoff-James orthogonality techniques. As an application of the results, we obtained a useful characterization ofbounded linear operators on a real Banach spaces. Also, using approximate 𝜖-Birkhoff -James orthogonality proved that a Banach space is a reflexive if and only if for any closed hyperspace of , there exists a rank one linear operator such that , for some vectors in and such that 𝜖 .Mathematics subject classification (2010): 46B20, 46B04, 47L05.

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