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 $A^*$ is nonempty. This extends several recent related results.