Operator norms of words formed from positive-definite matrices

Abstract

Let α₁,α₂,...,α_n,β₁,β₂,...,β_n be strictly positive reals with α₁ +α₂ +···+α_n = β₁ + β₂ +···+ β_n = s. In this paper, the inequality

|||A^α1B^β1A^α2 ···A^αnB^βn|||≤|||AB|||^s

when A and B are positive-definite matrices is studied. Related questions are also studied.