One criterion of semigroup product convergence

Abstract

Let exp(-tA) and exp(-tB) be C0 contraction semigroups on both K and ℬ, where K is a Hilbert space and ℬ is a reflexive Banach space such that the linear space ℬ∩K is dense both in K and ℬ. Let ℬ* be a dual pair of Banach spaces. In this paper we study some properties of infinitesimal operators of these semigroups. We show that under suitable assumptions there is some connection between the form-sum A+B and a closure of A1+B1, where -A1 is an infinitesimal operator of C0 contraction semigroup exp(-tA1) which is the extension by continuity on ℬ of C0 contraction semigroup exp(-tA)↾ ℬ ∩Kin ℬ. In particular we give some criterion of an m-accretive closability A1+B1 which may be applied for example to the Schrodinger operators acting in suitable Lp-spaces. Also this criterion together with ℬ properties of semigroups under consideration results in the establishment of the Lie-Trotter formulae.