Strong hypercontractivity in non-commutative holomorphic spaces

摘要

We introduce holomorphic algebras $H_q$ in the context of the q-Gaussianalgebra $\Gamma_q$ of Bozejko, K\"ummerer, and Speicher, and give aq-Segal-Bargmann transform for them. We then prove a strong hypercontractivitytheorem, generalizing Janson's strong (holomorphic) hypercontractivity, from$L^2(H_q) \to L^r(H_q)$ for r an even integer.