We derive estimates for approximation numbers of bounded linear operators between normed linear spaces. As special cases of our general results, approximation numbers of some weighted shift operators on p and those of isometries andprojections of norm 1 are found. In the case of finite-rank operators, we obtain estimatesfor the smallest nonzero approximation number in terms of their generalizedinverses. Also, we prove some results regarding the relation between approximationnumbers and the closedness of the range of an operator.We recall that the closednessof the range is a necessary condition for the boundedness of a generalized inverse.We give examples illustrating the results and also show that certain inequalities neednot hold. © Springer India 2013.

Thamban Nair M

Department of Mathematics

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Generalized Inverses and Approximation Numbers

Combinatorial Matrix Theory and Generalized Inverses of Matrices

In [1], Böttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H,{ej}j=1∞ is an orthonormal basis of H and Pn is the orthogonal projection onto the span of {ej}j=1n, then for each k ∈ Ndbl;, the sequence {sk(pnTPn)} converges to sk(T), where for a bounded operator A on H, sk(A) denotes the kth approximation number of A, that is, sk(A) is the distance from A to the set of all bounded linear operators of rank at most k - 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {Pn} and {Qn} are sequences of bounded linear operators on X and Y, respectively, such that for all {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 for all n ∈ ℕ and {QnTPn} converges to T under the weak operator topology, then {sk(QnTPn)} converges to sk(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of sk(QnTPn) to sk(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {Pn} and {Qn} on X and Y respectively such that (i) QnTPn is compact, (ii) {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 and (iii) {QnTPn} converges to T in the weak operator topology, then {sk(QnTPn)} converges to sk(T) if and only if sk(T) = sk(T′). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces. © 2009 Birkhäuser Verlag Basel/Switzerland.

Integral Equations and Operator Theory

Approximation numbers of operators on normed linear spaces

Bulletin of the London Mathematical Society

ON THE APPROXIMATION NUMBERS FOR THE FINITE SECTIONS OF BLOCK TOEPLITZ MATRICES

Periodica Mathematica Hungarica

Projections of normed linear spaces with closed subspaces of finite codimension as kernels

Introduction to Large Truncated Toeplitz Matrices Universitext

Block Toeplitz Matrices

Entropy, Compactness and the Approximation of Operators

Generalized inverses and approximation numbers

Journal	Data powered by TypesetCombinatorial Matrix Theory and Generalized Inverses of Matrices
Publisher	Data powered by TypesetSpringer India
Open Access	No