The case of equality in Hölder's inequality for operators and matrices

Abstract

Let p > 1 and 1/p + 1/q = 1. Consider Hölder’s inequality ∥ab∗∥1≤∥a∥p∥b∥q for the p-norms of n × n matrices. This note contains a simple proof (based on the case p = 2) of the fact that equality holds if and only if |a| p = λ|b| q for some λ ≥ 0. Without modification, the method of proof holds if a, b are matrices, compact operators, elements of a finite C∗ -algebra or a semi-finite von Neumann algebra.