Trotter-type formula for operator semigroups on product spaces

TL;DR

This paper introduces a Trotter-type product formula for approximating solutions of linear abstract Cauchy problems on product Banach spaces, achieving near-optimal convergence rates under certain conditions.

Contribution

It extends classical Trotter formulas by freezing components in product spaces and establishes convergence rates with bounded off-diagonal operators and holomorphic semigroups.

Findings

Convergence in the strong topology is achieved with stability estimates.

Operator norm convergence is established with near-optimal rates.

Main result applies when the dominant operator is holomorphic and coupling operators are bounded.

Abstract

We consider a Trotter-type-product formula for approximating the solution of a linear abstract Cauchy problem (given by a strongly continuous semigroup), where the underlying Banach space is a product of two spaces. In contrast to the classical Trotter-product formula, the approximation is given by freezing subsequently the components of each subspace. After deriving necessary stability estimates for the approximation, which immediately provide convergence in the natural strong topology, we investigate convergence in the operator norm. The main result shows that an almost optimal convergence rate can be established if the dominant operator generates a holomorphic semigroup and the off-diagonal coupling operators are bounded.