Some Remarks on Operator-Norm Convergence of the Trotter Product Formula on Banach Spaces

✍ Scribed by Sönke Blunck

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 206 KB
• Volume: 195
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2002.3965

✦ Synopsis

We extend the Trotter-Kato-Chernoff theory of strong approximation of C 0 semigroups on Banach spaces to operator-norm approximation of analytic semigroups with error estimate. As application we obtain a criterion for the operator-norm convergence of the Trotter product formula on Banach spaces with error estimate n À1 log n; provided one of the generators has a bounded H 1 functional calculus. For both results, we present versions for approximation in operator-ideal-norms such as the trace norm or the Hilbert-Schmidt norm. Finally, we give some remarks on the operator-norm convergence of the Trotter product formula for semigroups acting on a scale of L p -spaces.