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 \(K_2\).
Highlights
The aim of the paper is to present a new generalised proof of approximation theory developed in [6, 7]
Instead of a double-iteration procedure of [7] we extend in this paper the Chernoff approximation formula [5] and the Trotter–Neveu–Kato approximation theorem [8], Theorem IX.2.16, to the operator-norm topology
We follow here the idea of lifting the strongly convergent Chernoff approximation formula to operator-norm convergence [9, 11], whereas majority of results concerning this formula are about the strong operator topology, see, for example, review [2]
Summary
The aim of the paper is to present a new generalised proof of approximation theory developed in [6, 7]. For self-adjoint Trotter–Kato product formulae it allows to obtain optimal estimate for the rate of convergence in operator norm for Kato functions of class Kβ, where β = 2 (see [7]). Zagrebnov book [1], where different aspects of semigroup convergence in the strong operator topology are presented in great details. By Definition 1.1 and by the spectral theorem one gets that for any non-negative self-adjoint operator A the bounded operator-valued function t → f (tA) ∈ L(H) is strongly continuous in R+ and right-continuous on R+0 = R+ ∪ {0}, that is, s-limt→+0 f (tA) = 1. One of the main corollaries of the semigroup approximation results established in the present paper (Theorem 4.5) is the statement about operator norm convergence of the Trotter–Kato product formulae, see Section 5.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
