Sharp uniform convexity and smoothness inequalities for trace norms

Abstract

Summary We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsC _p , which are the analogs of the Lebesgue spacesL _p in non-commutative integration. The inequalities are all precise analogs of results which had been known inL _p , but were only known inC _p for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forC _p we obtain new and simple proofs of the known inequalities forL _p .