Schur multiplier norms for Loewner matrices

TL;DR

This paper investigates bounds on the Schur multiplier norms of Loewner matrices for various classes of functions, providing insights into the behavior of commutators and Schatten norms in operator theory.

Contribution

It introduces new upper bounds on Schur multiplier norms for Loewner matrices associated with concave, convex, and operator monotone functions, enhancing understanding of operator commutator norms.

Findings

Derived upper bounds for Schur multiplier norms for concave and convex functions.

Established bounds on the ratio of Schatten q-norms of commutators.

Obtained sharper bounds for operator monotone functions.

Abstract

We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten $q$-norms of commutators $||\;[A,f(B)]\;||_q/||\;[A,B]\;||_q$. We also consider operator monotone functions, for which sharper bounds are obtained.