A factorization of extremal extensions with applications to block operator matrices
type: Item
instance of: scholarly article
title: A factorization of extremal extensions with applications to block operator matrices
zbMATH Open document ID: 1223.47025
author: Maria Magdalena Nafalska
author: K.-H. Förster
published in: Acta Mathematica Hungarica
publication date: 2011-02-25T00:00:00Z
review text: Let \(A\) be a densely defined nonnegative operator in a separable Hilbert space \(\mathcal H\) with inner product \(\langle\cdot, \cdot\rangle\). There exist two nonnegative selfadjoint extensions, the Friedrichs extension \(A_{F}\) and the Kreĭn-von Neumann extension \(A_{N}\), with the property that every nonnegative selfadjoint extension \(\widetilde{A}\) of \(A\) satisfies the inequalities \[ \langle(A_{F}+\lambda)^{-1}f,f\rangle\leq \langle(\widetilde{A}+\lambda)^{-1}f,f\rangle \leq \langle(A_{N}+\lambda)^{-1}f,f\rangle\quad \text{for}\quad f\in {\mathcal H},\;\lambda\geq 0. \] A nonnegative selfadjoint extension \(\widetilde{A}\) of \(A\) is called extremal if \(\inf \{\langle\widetilde{A} (\varphi-f),\varphi-f\rangle: f\in{\mathcal D}(A)\}=0\) for all \(\varphi\in{\mathcal D}(\widetilde{A})\). The description of extremal extensions may be based on a special factorization. Define the operators \(Q\) and \(J\) by \(Q : {\mathcal H} \supset {\mathcal D}(A) \to {\mathcal H}_A\), \(f\mapsto Af\), \(J:{\mathcal H}_A \supset {\mathcal R}(A) \to {\mathcal H}\), \(Af \mapsto Af\), where \({\mathcal H}_A = \{ {\mathcal R}(A), \langle\cdot,\cdot\rangle\}\widehat{\;}\) is the completion of the range of \(A\) with respect to the inner product \(\langle Af, Ag\rangle\). In this case, we have the following factorization [see \textit{Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyén} and \textit{H. S. V. de Snoo}, Oper. Theory, Adv. Appl. 127, 41--81 (2001; Zbl 0996.47029)]: \[ A=JQ, \quad A_F=Q^\star Q^{\star\star}, \quad A_N=J^{\star\star}J^\star. \] In the paper under review, the authors consider two Hilbert spaces \(\mathcal H, \mathcal K\) and a nonnegative operator \(A = KC\) in \(\mathcal H\), where \(K\) and \(C\) are acting from \(\mathcal K\) into \(\mathcal H\) and from \(\mathcal H\) into \(\mathcal K\), respectively. It is not assumed that the operators \(A\), \(C\) and \(K\) are closable or densely defined. A~description of all extremal extensions by means of the facto…
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 5857786
zbMATH Keywords: Friedrichs extension
zbMATH Keywords: Kreĭn-von Neumann extension
zbMATH Keywords: extremal extension
zbMATH Keywords: factorisation
zbMATH Keywords: block operator matrix
MaRDI profile type: Publication
full work available at URL: s10474-010-9248-x
OpenAlex ID: W2028739976
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DOI: 10.1007/S10474-010-9248-X
description: scientific article; zbMATH DE number 5857786