Three observations regarding Schatten $p$ classes

Abstract

The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation of a new complemented subspace of C-p in the reflexive range (and p not equal 2). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of C-p in C-p(n) with small n and shows that l(p)(k) nicely embeds into C-n(p) only if n is at least proportional to k (and then of course the dimension of C-p(n) is at least of order k(2)). The third result concerns single elements of C-p(n) and shows that for p > 2 any n x n matrix of C-p norm one and zero diagonal admits, for every epsilon > 0, a k-paving of C-p norm at most a with k depending on epsilon and p only.