Kato's equality and spectral decomposition for positive C0-groups

Abstract

Let A be the generator of a C0-semigroup {T(t); t⩾0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies = (x∈D(A), x′ ∈D(A′)) (where q: E→E” is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.