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

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 \(\nu \rightarrow \infty \) .