An operator-coefficients free Poincar\'e inequality

TL;DR

This paper establishes a new Poincaré inequality that does not depend on operator coefficients, providing a novel bound involving free probability concepts.

Contribution

It introduces an operator-coefficients free Poincaré inequality using the projective tensor norm, expanding the theoretical framework in free probability.

Findings

Proves a new operator-coefficients free Poincaré inequality.

Utilizes the projective tensor norm in the inequality.

Provides a bound relating function variance to free derivatives.

Abstract

We prove the following operator-coefficients free Poincar\'{e} inequality: $$|f(X)-E[f(X)]|_{2}\leq2|X|_{2}\|\widehat{\partial}_{X:B}[f(X)]\|_{\pi}, \quad f(X) \in \mathrm{dom}(\widehat{\partial}_{X:B}),$$ where $\|\cdot\|_{\pi}$ is the projective tensor norm.