CLOSED IDEALS IN LIPSCHITZ ALGEBRAS AND HILBERT HARMONIC FORMS WITH LIVSIC THEOREM

Abstract

We characterize the closed ideals in some algebras of analytic functions, the algebra of absolutely convergent Taylor series and in analytic weighted Lipschitz algebras. The Hodge de Rham decomposition for anL^2 space of differential two-forms and vanishing of L^2 harmonic one-forms on path spaces are considered. We show the vanishing theorem of the Hodge-Kodaira operator and the generalized Clarc-Ocone formulae for differential forms. Some applications of Krien’s theory of regular symmetric operators to sampling and analytic sampling theory in a Hilbert space are investigated.