Dirichlet operators and the positive maximum principle

Abstract

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p≥1 andm not necessarily σ-finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), q≥p. It turns out that, in the limitq→∞,A satisfies the positive maximum principle. If the test functionsC ^∞ _c ⊂D(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 onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.