Trotter‐type formula for operator semigroups on product spaces

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 subsequently freezing 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 a convergence rate of (Formula presented.) can be established if the dominant operator generates a holomorphic semigroup and the off-diagonal coupling operators are bounded. The result is illustrated with examples, and also unbounded couplings are discussed.