Approximation Numbers of Operators on Normed Linear Spaces

摘要

In [1], Bottcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H, {ej}(j=1)(infinity) is an orthonormal basis of H and P-n is the orthogonal projection onto the span of {ej}(j=1)(n) = 1, then for each k. N, the sequence {s(k)(PnTPn)} converges to s(k)(T), where for a bounded operator A on H, sk(A) denotes the kth approximation number of A, that is, s(k)(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 {P-n} and {Q(n)} are sequences of bounded linear operators on X and Y, respectively, such that parallel to P-n parallel to parallel to Q(n)parallel to <= 1 for all n is an element of N and {Q(n)TP(n)} converges to T under the weak operator topology, then {sk(Q(n)TP(n))} 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(Q(n)TP(n)) 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 {P-n} and {Q(n)} on X and Y respectively such that (i) Q(n)TP(n) is compact, (ii) parallel to P-n parallel to parallel to Q(n)parallel to = 1 and (iii) {Q(n)TP(n)} converges to T in the weak operator topology, then {sk(Q(n)TP(n))} 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.