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Trotter-Kato Product Formulae: Operator-Norm Topology and Error Bounds

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Trotter-Kato Product Formulæ

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 296))

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Abstract

In this chapter, we present two fundamental results about the operator-norm optimal error bound estimates for convergence rate of the Trotter-Kato product formul for semigroups with noncommuting self-adjoint generators.

The ITTZ-Theorem (Sect. 7.1) shows that self-adjointness of the sum of a couple of generators, is a crucial condition for the operator-norm convergence of the Trotter-Kato product formul with the optimal error bound estimate for the rate of convergence.

The INZ-Theorem (Sect. 7.4) concerns the case when it is the form-sum of a couple of generators is self-adjoint. In Sect. 7.4 we establish dependence of the optimal operator-norm error bound estimate for convergence rate of the Trotter-Kato product formul on relations between fractional powers of the self-adjoint generators.

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Authors and Affiliations

  1. Université d’Aix-Marseille - CNRS, Institute de Mathématiques de Marseille, Marseille Cedex 13, France

    Valentin A. Zagrebnov

  2. Berlin, Germany

    Hagen Neidhardt

  3. Department of Mathematics, Kanazawa University, Kanazawa, Japan

    Takashi Ichinose

Authors
  1. Valentin A. Zagrebnov
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  2. Hagen Neidhardt
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  3. Takashi Ichinose
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Zagrebnov, V.A., Neidhardt, H., Ichinose, T. (2024). Trotter-Kato Product Formulae: Operator-Norm Topology and Error Bounds. In: Trotter-Kato Product Formulæ. Operator Theory: Advances and Applications, vol 296. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-56720-9_7

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