Spectral representations of unbounded non-linear operators on Hilbert space
type: Item
instance of: scholarly article
title: Spectral representations of unbounded non-linear operators on Hilbert space
zbMATH Open document ID: 0542.47023
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let \({\mathcal K}\) be a separable complex \(\infty\)-dimensional Hilbert space and let \({\mathcal F}\) be the Fock space of symmetric tensors over \({\mathcal K}\). We consider non-linear operators T from \({\mathcal K}\) to \({\mathcal F}\) defined on a dense subspace \({\mathcal D}\) in \({\mathcal K}\) with range in \({\mathcal F}\). A symmetry and reality condition is imposed on the operators T under consideration. They are generally unbounded and have different extensions \(\tilde T\) defined on subspaces \(\tilde {\mathcal D}\) in \({\mathcal K}\) containing \({\mathcal D}\). Generalizing a result of Arveson for bounded operators (alias functions from \({\mathcal K}\) to \({\mathcal F})\), we show that if T is affiliated with a maximal abelian von Neumann algebra in \(B({\mathcal K})\), then it follows that there is an extension\~T of T which is unitarily equivalent to a (non-linear) multiplication operator.
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 3861851
zbMATH Keywords: Fock space of symmetric tensors
zbMATH Keywords: symmetry and reality condition
zbMATH Keywords: multiplication operator
author: Palle Jorgensen
MaRDI profile type: Publication
full work available at URL: pjm.1984.111.93
OpenAlex ID: W1977532580
DOI: 10.2140/PJM.1984.111.93
description: scientific article; zbMATH DE number 3861851