Wehrl-type inequalities for Bergman spaces on domains in \(\mathbb{C}^d\) and completely positive maps (Q6616794)

Let \(H\) be a subspace of some Lebesgue space \(L^2(\Omega,d\sigma)\) which possesses a reproducing kernel \(K(z,w)\). By the reproducing property and Cauchy-Schwarz inequality, the multiplication operator \(T: f(z)\mapsto K(z,z)^{-1/2}f(z)\) satisfies \N\[ |Tf(z)| \le \|f\|_H, \qquad\forall f\in H. \] \NIn other words, \N\[ \|Tf\|_{L^2(\Omega, K(z,z)\,d\sigma(z))} = \|f\|_H, \qquad \|Tf\|_{L^\infty (\Omega, K(z,z)\,d\sigma(z))} \le \|f\|_H . \] \NIt follows by interpolation that \(T\) is a contraction from \(H\) into \(L^p(\Omega, K(z,z)\,d\sigma(z))\) for any \(p\ge2\). For \(H\) the appropriately weighted Bergman space of holomorphic functions on the disc or the Fock space of entire functions on \(\mathbf C^n\), this \(L^2\)-\(L^p\) contractivity is precisely the content of the (generalized) Wehrl inequalities of \textit{E. H. Lieb} and \textit{J. P. Solovej} [Acta Math. 212, No. 2, 379--398 (2014; Zbl 1298.81116)], who gave a proof for \(p\) an even integer and further showed that equality takes place only on scalar multiples of reproducing kernels. Alternatively, the contractive property of \(T\) in those settings can be interpreted as a certain \(L^2\)-\(L^p\) inequality for matrix coefficients of representations of \(SU(n)\), \(SU(1,1)\) or the \(AX+B\) group, respectively.\N\NIn the paper under review, the above Wehrl inequalities are extended, for \(p\) an even integer, to the setting of weighted Bergman spaces on a bounded symmetric domain \(\Omega=G/K\) in \(\mathbf C^n\). The proof uses an argument completely different from those of Lieb and Solovej, namely a simple observation concerning reproducing kernels of tensor products of weighted Bergman spaces. Furthermore, these inequalities are shown to relate to -- or, rather, be special cases of -- certain inequalities concerning the Berezin transform associated to the above-mentioned weighted Bergman spaces. Finally, the whole topic is put into the context of \(G\)-invariant completely positive maps from operators on one weighted Bergman space into operators on another such space, where the \(L^p\) norm of a Bergman space function \(f\) is shown to be equal to an appropriate ``semiclassical'' limit of the trace of the images of the associated rank-one projection \(f\otimes f^*\).\N\NFor the entire collection see [Zbl 1537.32001].