Weil representation and norms of Gaussian operators
Let \({\mathcal H}\) be a Hilbert space and \(\Gamma ({\mathcal H})\) the semigroup of contraction operators on \({\mathcal H}\) equipped with the weak operator topology. Then \(\Gamma ({\mathcal H})\) is a compact semitopological semigroup (i.e. multiplication is separately continuous). If \(G\) is a group and \(T\) a unitary representation of \(G\) in \({\mathcal H}\), then the closure of \(T(G)\) in \(\Gamma ({\mathcal H})\) is a compactification of \(G\) which has proved especially useful in the case where \(G\) is the inductive limit of classical groups. In the paper under review the author considers the inductive limit of the metaplectic groups \(\text{Mp} (n, \mathbb{R})\) and their Shale-Segal-Weil representations which are realized on the Fock space of holomorphic maps on \(\mathbb{C}^n\) which are square integrable with respect to a suitable Gaussian measure. Of particular importance in this context are Gaussian operators, i.e., integral operators with Gaussian kernels. The main result of the paper is an explicit formula for the norm of such Gaussian operators. Further the author discusses various open problems arising in this context.
scientific article; zbMATH DE number 859667