Remarks on the Operator-Norm Convergence of the Trotter Product Formula

Remarks on the Operator-Norm Convergence of the Trotter Product Formula We revise the operator-norm convergence of the Trotter product formula for a pair $$\{A,B\}$$ { A , B } of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

Remarks on the Operator-Norm Convergence of the Trotter Product Formula

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-018-2424-z
Publisher site
See Article on Publisher Site

Abstract

We revise the operator-norm convergence of the Trotter product formula for a pair $$\{A,B\}$$ { A , B } of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.

Journal

Integral Equations and Operator TheorySpringer Journals

Published: Mar 12, 2018

References

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