Commutator inequalities via Schur products

TL;DR

This paper develops inequalities for commutators involving functions of unbounded operators using Schur products and matrix approximations, extending classical results to a broader operator setting.

Contribution

It introduces a novel approach to bound commutators of functions of unbounded operators via Schur product techniques in an infinite matrix framework.

Findings

Established inequalities for commutators involving functions of unbounded operators.

Applied Bennett's classical Schur product inequality to operator commutator bounds.

Provided a new method for approximating commutators using matrix Schur products.

Abstract

For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some complex Borel functions g(t) we establish inequalities of the type ||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D,y] and a scalar matrix. A classical inequality of Bennett on the norm of Schur products may then be applied to obtain the results.