Approximations of self-adjoint \(C_0\)-semigroups in the operator-norm topology

Valentin Zagrebnov

Abstract


The paper improves approximation theory based on the Trotter–Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter–Kato product formulae for Kato functions from the class \(K_2\).

Keywords


Strongly continuous semigroup; Chernoff approximation formula; Trotter–Kato product formulae

Full Text:

PDF

References


Bobrowski, A., Convergence of One-parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere, New Mathematical Monographs 30, Cambridge University Press, Cambridge, 2016.

Butko, Ya. A., The method of Chernoff approximation, To appear in: Semigroups of Operators: Theory and Applications SOTA-2018, Springer Proceedings in Mathematics, 2020.

Cachia, V., Zagrebnov, V. A., Operator-norm convergence of the Trotter product formula for holomorphic semigroups, J. Oper. Theory 46 (2001), 199–213.

Cachia, V., Zagrebnov, V. A., Operator-norm approximation of semigroups by quasisectorial contractions, J. Funct. Anal. 180 (2001), 176–194.

Chernoff, P. R., Note on product formulas for operator semigroups, J. Funct. Anal. 2 (1968), 238–242.

Ichinose, T., Tamura, H., The norm convergence of the Trotter–Kato product formula with error bound, Commun. Math. Phys. 217 (2001), 489–502.

Ichinose, T., Tamura, Hideo, Tamura, Hiroshi, Zagrebnov, V. A., Note on the paper “The norm convergence of the Trotter–Kato product formula with error bound” by Ichinose and Tamura, Commun. Math. Phys. 221 (2001), 499–510.

Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980, Corrected Printing of the Second Edition.

Neidhardt, H., Zagrebnov, V. A., Trotter–Kato product formula and symmetrically normed ideals, J. Funct. Anal. 167 (1999), 113–167.

Zagrebnov, V. A., Quasi-sectorial contractions, J. Funct. Anal. 254 (2008), 2503–2511.

Zagrebnov, V. A., Comments on the Chernoff (sqrt{n})-lemma, in: Functional Analysis and Operator Theory for Quantum Physics (The Pavel Exner Anniversary Volume), European Mathematical Society, Z¨urich, 2017, 565–573.

Zagrebnov, V. A., Gibbs Semigroups, Operator Theory Series: Advances and Applications 273, Bikhauser-Springer, Basel 2019.




DOI: http://dx.doi.org/10.17951/a.2019.73.2.175-194
Data publikacji: 2020-01-16 07:29:37
Data złożenia artykułu: 2020-01-04 21:27:26


Statistics

Total abstract view - 394
Downloads (from 2020-06-17) - PDF - 238

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 Valentin Zagrebnov