Trotter-Kato product formula and operator-norm convergence
type: Item
instance of: scholarly article
title: Trotter-Kato product formula and operator-norm convergence
zbMATH Open document ID: 0949.47019
published in: Communications in Mathematical Physics
publication date: 1999-12-16T00:00:00Z
review text: For two non-negative self-adjoint operator \(A\) and \(B\) on a separable Hilbert space \(\mathcal{H}\) T. Kato showed the convergence of the Trotter products \[ \text{strong}- \lim_{n\rightarrow \infty} \bigl( f(tA/n) g(tB/n) \bigr)^n = e^{-tH} P_0 \] to the semigroup generated by the form sum \[ H=A\dot{+} B \quad \text{on}\quad {\mathcal{H}}_0:=\overline{D(A^{1/2})\cap D(B^{1/2})} \] with \(P_0\) the orthogonal projection from \(\mathcal{H}\) onto \({\mathcal{H}}_0\) and for Kato-functions \(f,g\). In this paper the authors investigate the operator norm convergence (away from zero) of these products. Therefore, they establish necessary and sufficient conditions for this convergence including a resolvent convergence in operator norm. They obtain the operator norm convergence of the products away from zero for every regular Kato--function \(f\) (such as \(e^{-t}\)) and every Kato--function \(g\) if \(f(t_0 A)\) is compact for some \(t_0>0\). This condition holds if the resolvent of one of the underlying operators is compact. The compactness of the resolvent turns out to be also a sufficient condition for the norm convergence of the Trotter products. On the other side, for the case \(D(A^{1/2}) \subseteq D(B^{1/2})\) they obtain the same convergence of the Trotter products if \(A^{1/2}\) is relatively compact to \(B^{1/2}\), i.e. \(B^{1/2}A^{-1/2}\) is compact.
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 1379909
zbMATH Keywords: Trotter product formula
zbMATH Keywords: operator-norm convergence
zbMATH Keywords: Kato function
zbMATH Keywords: compactness of the resolvent
author: Hagen Neidhardt
author: V. A. Zagrebnov
reviewed by: Rainer Nagel
MaRDI profile type: Publication
full work available at URL: s002200050671
OpenAlex ID: W2004603757
DOI: 10.1007/S002200050671
description: scientific article; zbMATH DE number 1379909