11 Entropy numbers versus s-numbers

The Gelfand and the Kolmogorov numbers detect compactness on every Banach space. The approximation numbers do so only under an extra hypothesis: every space without the approximation property is the target of a compact operator that is not a limit of finite-rank operators (Enflo). The equivalence \(a_n(S) \to 0 \iff S\) compact holds for operators into spaces with the approximation property, in particular between Hilbert spaces, where all s-numbers agree with \(a_n\).

This chapter compares \(c_n\), \(d_n\) and \(h_n\) with the entropy numbers of Chapter 10, in both directions: upper bounds for \(c_n, d_n\) in terms of \(e_n\) give “compact \(\Rightarrow \) decay”, and a lower bound for \(e_n\) gives the sharp comparison. Through Corollary 7.14 the bound for \(h_n\) extends to every s-number sequence (Section 11.5). Throughout this chapter \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \).