The von Neumann problem for nonnegative symmetric operators
type: Item
instance of: scholarly article
title: The von Neumann problem for nonnegative symmetric operators
zbMATH Open document ID: 1082.47018
published in: Integral Equations and Operator Theory
publication date: 2005-04-29T00:00:00Z
review text: \textit{J. von Neumann} [Math. Ann. 102, 49--131 (1929; JFM 55.0824.02)] proposed a problem about the existence and the description of all self-adjoint extensions preserving the lower bound of a given densely defined symmetric operator which is bounded from below. It is the aim of the paper under review to present a new operator-valued parametrization of the domains and associated closed forms of all nonnegative self-adjoint extensions (among them the minimal Kreĭn--von Neumann extension) of a densely defined, closed nonnegative symmetric operator with, in general, zero lower bound. The approach considered by the present authors, based on the Friedrichs extension, is entirely new and provides a complete solution to von Neumann's problem in an explicit form. In this context, \textit{M.~S.~Birman}'s description of all nonnegative self-adjoint extensions of a positive definite symmetric operator [Mat. Sb. 38(80), 431--450 (1956; Zbl 0070.12102)] as well as Kreĭn's resolvent formula about canonical and generalized resolvents of a symmetric operator are obtained as consequences. An application to the description of all nonnegative quantum Hamiltonians in the point-interaction model with a finite number of point-interactions in \({\mathbb R}^3\) concludes the paper.
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 2162375
zbMATH Keywords: nonnegative self-adjoint extension
zbMATH Keywords: Friedrichs and Kreĭn--von Neumann extension
zbMATH Keywords: Kreĭn--Langer \(Q\)-function
zbMATH Keywords: boundary triplet
zbMATH Keywords: Weyl--Titchmarsh operator-valued function
author: Yu M. Arlinskii
author: Eduard Tsekanovskii
MaRDI profile type: Publication
full work available at URL: s00020-003-1260-x
OpenAlex ID: W1996619495
DOI: 10.1007/S00020-003-1260-X
description: scientific article; zbMATH DE number 2162375