Commutator conditions implying the convergence of the Lie–Trotter products

Abstract

Abstract. In this paper we investigate commutator conditions for two strongly continuous semigroups (T (t)) t≥0 and (S(t)) t≥0 on a Banach space implying the convergence of the Lie-Trotter products [T (

Lie-Trotter product formula for locally equicontinuous and tight Markov semigroup

“…Hence, a different approach is needed. The analysis of commutator type conditions as in [21,10] avoids considering generators and their domains and may be easier to verify.…”

“…Our starting point is the conditions for convergence of the Lie-Trotter product formula formulated by Kühnemund and Wacker in [21]. This result appears to be a very useful tool in proving the convergence of the Lie-Trotter scheme without the need to have knowledge about generators of the semigroups involved.…”

“…We extend Kühnemund and Wacker's case to semigroups of Markov operators on spaces of measures and present weaker sufficient conditions for convergence of the switching scheme. Our method of proof builds on [21], while the specific commutator condition that we employ (Assumption 3) is motivated by [10].…”

“…In Section 5 we prove the convergence of the Lie-Trotter product formula for Markov operators. We provide more general assumptions then those provided in the Kühnemund-Wacker paper (see [21]). As our semigroups are not strongly continuous and usually not bounded, we use the concept of (local) equicontinuity (see e.g.…”

On the Operator Splitting Method: Nonlinear Balance Laws and a Generalization of Trotter-Kato Formulas

“…We point out that condition (4.2) holds for any ω(t) ≤ t α , with α > 0. Hence, the bound on the commutator required by Proposition 4.4 is weaker than the one required in [19] in the linear case.…”