Free Stein kernels and an improvement of the free logarithmic Sobolev inequality

TL;DR

This paper introduces a free Stein kernel relative to a semicircular law, leading to an improved free logarithmic Sobolev inequality and providing convergence rates in the free CLT.

Contribution

It develops a free Stein kernel concept, enhancing the free logarithmic Sobolev inequality and establishing convergence rates in the free Central Limit Theorem.

Findings

Derived a free Stein kernel for various operator families.

Established an improved free HSI inequality.

Provided convergence rates in the multivariate free CLT.

Abstract

We introduce a free version of the Stein kernel, relative to a semicircular law. We use it to obtain a free counterpart of the HSI inequality of Ledoux, Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev inequality of Biane and Speicher, as well as a rate of convergence in the (multivariate) entropic free Central Limit Theorem. We also compute the free Stein kernels for several relevant families of self-adjoint operators.