Operator-norm Trotter product formula on Banach spaces

TL;DR

This paper reviews results on the operator-norm convergence of the Trotter product formula for semigroups in Banach spaces, highlighting developments from strong to operator-norm convergence over time.

Contribution

It consolidates known results on operator-norm convergence of the Trotter product formula in Banach spaces, emphasizing recent advancements.

Findings

Operator-norm convergence established in Banach spaces in 2001.

Strong convergence known since 1959, extended to Hilbert spaces in 1993.

The paper collects and discusses these convergence results.

Abstract

In this paper we collect results concerning the {operator-norm} convergent {Trotter} product formula for two semigroups $\{\e^{- t A}\}_{t\geq 0}$, $\{\e^{- t B}\}_{t\geq 0}$, with densely defined generators $A$ and $B$ in a {Banach} space. Although the {strong} convergence in Banach space for contraction semigroups is known since the seminal paper by Trotter (1959), which after more than three decades was extended to convergence in the {operator-norm} topology in {Hilbert} spaces by Rogava (1993), the {operator-norm} convergence in a {Banach} space was established only in (2001).