Approximation Numbers of Operators on Normed Linear Spaces

Abstract

In [1], Bottcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space \(H, \{e_{j}\}_{j=1}^{\infty}\) is an orthonormal basis of H and P n is the orthogonal projection onto the span of \(\{e_{j}\}_{j=1}^{n}\), then for each \(k \in {\mathbb{N}}\), the sequence \(\{s_{k}(P_{n}TP_{n})\}\) converges to s k (T), where for a bounded operator A on H, s k (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 \(\|P_n\| \|Q_n\| \leq 1 \) for all \(n \in {\mathbb{N}}\) and {Q n TP n } converges to T under the weak operator topology, then \(\{s_{k}(Q_{n}TP_{n})\}\) converges to s k (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 \(s_{k}(Q_{n}TP_{n})\) to s k (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) \(\|P_{n}\| \|Q_{n}\| \leq 1 \) and (iii) \(\{Q_{n}TP_{n}\}\) converges to T in the weak operator topology, then \(\{s_k(Q_{n}TP_{n})\}\) converges to s k (T) if and only if \(s_{k}(T) = s_{k}(T^\prime)\). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.