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Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

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Analysis and Operator Theory

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 146))

Abstract

The paper is devoted to evolution equations of the form

$$\begin{aligned} \frac{\partial }{\partial t}u(t) = -(A + B(t))u(t), \quad t \in {\mathcal {I}}= [0,T], \end{aligned}$$

on separable Hilbert spaces where A is a non-negative self-adjoint operator and \(B(\cdot )\) is family of non-negative self-adjoint operators such that \(\mathrm {dom}(A^{\alpha }) \subseteq \mathrm {dom}(B(t))\) for some \({\alpha }\in [0,1)\) and the map \(A^{-{\alpha }}B(\cdot )A^{-{\alpha }}\) is Hölder continuous with the Hölder exponent \({\beta }\in (0,1)\). It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition \({\beta }> 2{\alpha }-1\) is satisfied. The convergence rate for the approximation is given by the Hölder exponent \({\beta }\). The result is proved using the evolution semigroup approach.

On the occasion of the 100th birthday of Tosio Kato

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Acknowledgements

We thank Takashi Ichinose and Hideo Tamura for the explanation of details of the proof of Theorem 1.1 of [14], which makes possible to prove Lemmas 3.8 and 3.9.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Hagen Neidhardt & Artur Stephan

  2. Institut de Mathématiques de Marseille (UMR 7373), Université d’Aix-Marseille, CMI - Technopôle Château-Gombert, 39 rue F. Joliot Curie, 13453, Marseille, France

    Valentin A. Zagrebnov

Authors
  1. Hagen Neidhardt
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  2. Artur Stephan
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  3. Valentin A. Zagrebnov
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Correspondence to Valentin A. Zagrebnov .

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Editors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Aix-Marseille University, Marseille, France

    Valentin A. Zagrebnov

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Neidhardt, H., Stephan, A., Zagrebnov, V.A. (2019). Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces. In: Rassias, T.M., Zagrebnov, V.A. (eds) Analysis and Operator Theory . Springer Optimization and Its Applications, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-030-12661-2_13

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