Dirichlet operators and the positive maximum principle
Integral Equations and Operator Theory, 41 (1):
74--92 (Mar 1, 2001)DOI: 10.1007/BF01202532
Abstract
Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onLp(m), p≥1 andm not necessarily $\sigma$-finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesLq(m), q≥p. It turns out that, in the limitq→∞,A satisfies the positive maximum principle. If the test functionsCc∞⊂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)) onLp(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.