Log-majorizations and norm inequalities for exponential operators

Abstract

Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii–Wielandt and the Gelfand– Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden–Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.