Abstract multiplication semigroups (Q1319335)

We characterize the linear operators on a Banach lattice which are generators of multiplication semigroups. The main result is as follows: Suppose \(A\) is a positive band preserving operator with \(D(A)\) a Riesz subspace which generates a dense ideal. Then \(-A\) admits a unique extension to a multiplication semigroup. If \(D(A)\) is a dense ideal, then this extension is the closure. We use this to prove the following: if \(A\) is a linear operator on \(E\) whose domain generates a dense ideal, then \(A\) generates a multiplication semigroup if and only if there is a real \(\lambda\in\rho(A)\) such that \(R(\lambda,A)= (\lambda- A)^{-1}\) is positive and band preserving. We also study the duality theory of multiplication semigroups.