L 1-spectrum of Banach space valued Ornstein–Uhlenbeck operators

Abstract

We characterize the L 1(E,μ ∞)-spectrum of the Ornstein–Uhlenbeck operator \(Lf(x)=\frac{1}{2}\mathop{\mathrm{Tr}}QD^{2}f(x)+\langle Ax,Df(x)\rangle\) , where μ ∞ is the invariant measure for the Ornstein–Uhlenbeck semigroup generated by L. The main result covers the general case of an infinite-dimensional Banach space E under the assumption that the point spectrum of A * is nonempty and extends several recent related results.