Strong hypercontractivity in non-commutative holomorphic spaces

Abstract

We introduce holomorphic algebras $H_q$ in the context of the q-Gaussian algebra ${\Gamma }_q$ of Bozejko, Kümmerer, and Speicher, and give a q-Segal-Bargmann transform for them. We then prove a strong hypercontractivity theorem, generalizing Janson's strong (holomorphic) hypercontractivity, from $L^2(H_q)\to L^r(H_q)$ for r an even integer.

25 pages, uses package xy

References in corpus (3)

• $q$-Gaussian processes: non-commutative and classical aspects
• Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration
• Gaussian Random Matrix Models for q-deformed Gaussian Random Variables

Cited by in corpus (10)

• Strong Haagerup inequalities for free R-diagonal elements
• Strong Haagerup inequality with operator coefficients
• Type B Gaussian Statistics as Noncommutative Central Limits
• Segal-Bargmann transform: the $q$-deformation
• Hypercontractivity on the $q$-Araki-Woods algebras
• Hypercontractivity for log-subharmonic functions
• R-diagonal dilation semigroups
• The $(q,t)$-Gaussian Process
• On the norm of the $q$-circular operator
• Central limit theorem associated to Gaussian operators of type B