s-numbers of projections in Banach spaces (Q1826089)

Lindenstrauss and Tzafriri proved that a Banach space X is isomorphic to a Hilbert space if and only if there is a \(C>0\) so that every finite dimensional subspace of X has a projection on it of norm \(\leq C\). Recently, Pisier has given an analogue of this result to the effect that a Banach space is K-convex if and only if there is a \(C>0\) such that every n-dimensional subspace has a projection P on it with \(e_ n(P)\leq C\) for all n, where \(\{e_ n(P)\}^{\infty}_{n=1}\) is the sequence of entropy numbers of P. This paper is a study of similar characterizations of certain types of Banach spaces in which the entropy numbers in the above result are replaced by some other set of s-numbers of projection P - e.g. Hilbert numbers, Weyl numbers, or approximation numbers.