Approximation of Modulus Semigroups and their Generators

Abstract

The central result of this paper is a sandwiching theorem for semigroups acting on Banach lattices with order continuous norm. As a preparation we show that the norm of a Banach lattice \(X\) is order continuous if and only if every order bounded weak null sequence in \(X_+\) is a norm null sequence. From the sandwiching result we deduce approximation formulas for the modulus semigroup and its generator. For example, if \(A\) generates a dominated \(C_0\)-semigroup \(T\) we show that \(t\mapsto(n/t|R(n/t,A)|)^n\) converges to the modulus semigroup of \(T\) as \(n\to\infty\), and \(\frac{1}{s}(|T(s)|-I)\) converges (in the strong resolvent sense) to the generator of the modulus semigroup of \(T\) as \(s\to 0\).