A remark on operator-norm convergence of Trotter-Kato product formula
type: Item
instance of: scholarly article
title: A remark on operator-norm convergence of Trotter-Kato product formula
zbMATH Open document ID: 0976.47022
DOI: 10.1007/BF01194484
author: Hiroshi Tamura
published in: Integral Equations and Operator Theory
publication date: 2002-01-06T00:00:00Z
review text: It has been shown by Neidhardt and Zagrebnov that if the selfjoint operators \(A\) and \(B\) on some Hilbert space \(H\) with domains \(D(A)\) and \(D(B)\), respectively, satisfy, for some \(\alpha\in ]1/2,1[\) and \(a\in ]0,1[\), (i) \(A\geq I\), \(B\geq 0\), (ii) \(D(A^\alpha)\subset D(B^\alpha)\), \(\|B^\alpha x\|\leq a\|A^\alpha x\|\) for all \(x\in D(A)\), and (iii) \(D((A+ B)^\alpha)\subset D(A^\alpha)\), where \(A_B\) is the form-sum, then there exists a constant \(c>0\) such that \[ \|\exp\{- t(A+B)\}-(\exp\{- tA/n\}\exp\{- tB/n\})^n\|\leq c/n^{2\alpha- 1} \] uniformly in \(t\geq 0\). The author constructs an example showing that the bound in this estimate is the best possible as far as the power of \(n\) is concerned.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 41A80
zbMATH DE Number: 1523222
zbMATH Keywords: Trotter-Kato product formula
zbMATH Keywords: rate of convergence
zbMATH Keywords: selfjoint operators
zbMATH Keywords: form-sum
reviewed by: Heinrich Hering
MaRDI profile type: Publication
cites work: Note on product formulas for operator semigroups
cites work: Error bounds on exponential product formulas for Schrödinger operators
cites work: Error estimate in operator norm for Trotter-Kato product formula
cites work: Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations
cites work: Q3910982
cites work: On error estimates for the Trotter-Kato product formula
cites work: Fractional powers of self-adjoint operators and Trotter-Kato product formula
cites work: On the Product of Semi-Groups of Operators
full work available at URL: https://doi.org/10.1007/bf01194484
OpenAlex ID: W2028314964
description: scientific article; zbMATH DE number 1523222