Non-symmetric translation invariant Dirichlet forms

Abstract

In order to treat certain "non-symmetric" potential theories, It6 I-4] and Bliedtner [1] have generalized Beurling and Deny's theory of Dirichlet spaces by replacing the inner product in the Dirichlet space with a bilinear form defined on a real, regular functional space. This bilinear form, called a Dirichlet form, is supposed to be continuous and coercive, and furthermore to satisfy a "contraction"-condition. When the underlying space is a locally compact abelian group, there is a complete characterization of translation invariant Dirichlet spaces in terms of real, negative definite functions on the dual group. The main purpose of the paper is to obtain an analogous characterization (Theorem 3.7) of translation invariant Dirichlet forms, in terms of complex, negative definite functions on the dual group. The contents of the paper may be summarized in the following way: In w 1 we study positive closed sesquilinear forms on an abstract complex Hilbert space, and using the Lax-Milgram theorem, it is shown, that with every such form is associated a resolvent. The theory is specialized in w 2 to sesquilinear forms on the Hilbert space L 2 (X, ~), where some relations with normal contractions are discussed. In particular an example of a positive closed form fl is given, such that the unit contraction operates with respect to fl, but not with respect to the form fl*, which is adjoint to ft. However, in the case of a translation invariant, positive closed form fl on L 2 (G), where G is a locally compact abelian group, it is shown in w 3, that the unit contraction operates with respect to fl if and only if it operates with respect to fl*, and we finish by establishing the above mentioned characterization of translation invariant Dirichlet forms.