WIAS Preprint No. 3030, (2023)

Trotter-type formula for operator semigroups on product spaces


  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2020 Mathematics Subject Classification

  • 47D06 15A60


  • Strongly continuous semigroups of linear operators, split-step method, Trotter-product formula, time discretization, product space, tensor space, block operator matrices, operator-norm convergence rate estimate, inhomogeneous abstract Cauchy problems




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.

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