Weil representation and norms of Gaussian operators
type: Item
instance of: scholarly article
title: Weil representation and norms of Gaussian operators
zbMATH Open document ID: 0848.22017
DOI: 10.1007/BF01079009
publication date: 1996-05-28T00:00:00Z
review text: 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.
reviewed by: Joachim Hilgert
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 859667
zbMATH Keywords: Hilbert space
zbMATH Keywords: contraction operators
zbMATH Keywords: weak operator topology
zbMATH Keywords: unitary representation
zbMATH Keywords: metaplectic groups
zbMATH Keywords: Shale-Segal-Weil representations
zbMATH Keywords: Fock space
zbMATH Keywords: Gaussian operators
author: G. I. Olshanskii
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01079009
OpenAlex ID: W2081304045
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 859667