Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL^p(m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL^q(m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC_cD(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL²(m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL²(m) with explicitly given Beurling-Deny formula.