From Matrix to Operator Inequalities

TL;DR

This paper extends Loewner's method to prove operator inequalities by reducing complex relations on bounded operators to matrix cases, enabling new results in noncommutative analysis.

Contribution

It introduces a meta-theorem that connects residually finite dimensional relations with operator inequalities, broadening the scope of matrix-based proofs.

Findings

Proves new operator inequalities using the meta-theorem.

Establishes bounds on norms of exponentials and commutators.

Provides applications to noncommutative *-polynomials.

Abstract

We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x \leq y on bounded operators is our model for a definition for C*-relations of being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved, and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators and "positive" noncommutative *-polynomials.