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# Approximation numbers of operators on normed linear spaces

Date Issued

01-12-2009

Author(s)

Abstract

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.

Volume

65