The strong Haagerup inequality for q-circular systems

TL;DR

This paper extends the strong Haagerup inequality to q-circular systems for |q|<1, generalizing free case results and applying to ultracontractivity of q-Ornstein--Uhlenbeck semigroup.

Contribution

It proves a new strong Haagerup inequality for q-circular systems, including cases with neither free independence nor -diagonality.

Findings

Established strong Haagerup inequality for q-circular systems with |q|<1

Proved ultracontractivity for q-Ornstein--Uhlenbeck semigroup

Derived sharp rates for Haagerup and ultracontractive inequalities

Abstract

Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with $\sqrt{n}$, rather than linearly in $n$, on homogeneous polynomials of degree $n$. In this paper, we prove a similar inequality for $q$-circular systems for $|q|<1$, generalizing the free case when $q=0$. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor $\mathscr{R}$-diagonality. As an application, we prove a strong ultracontractivity theorem for the $q$-Ornstein--Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.