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