Wehrl-Type Inequalities for Bergman Spaces on Domains in $\mathbb C^d$ and Completely Positive Maps

Abstract

We prove certain Wehrl type $L^p$-inequalities for the Bergman spaces on bounded domains D in $\mathbb C^d$ and apply the result to bounded symmetric domains $D=G/K$. We introduce also G-invariant completely positive trace preserving map $A\rightarrow \mathcal T(A)$ from operators A on a weighted Bergman space $H_\mu$ on $D=G/K$ to operators $\mathcal T(A)$ on another weighted Bergman space $H_{\mu +\nu }$ and prove that the $L^p$-norm of Bergman functions $f\in H_\mu$ can be obtained as limit of trace of $\mathcal T({f\otimes f^*})$ as the weight $\nu \rightarrow \infty$.