L1-spectrum of Banach space valued Ornstein-Uhlenbeck operators

Abstract

We characterize the $L^1(E;{\mu }_{\infty })$-spectrum of the Ornstein-Uhlenbeck operator, where ${\mu }_{\infty }$ is the invariant measure for the Ornstein-Uhlenbeck semigroup. The main result covers the general case of an infinite-dimensional Banach space E under the assumption that the point spectrum of drift operator $A^{\ast }$ is nonempty. This extends several recent related results.