SECOND QUANTIZATION AND THE Lp-SPECTRUM OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS

Abstract

The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric Ornstein–Uhlenbeck operator L associated with the infinite-dimensional Langevin equation [Formula: see text] where A is the generator of a strongly continuous semigroup on a Banach space E and W is a cylindrical Wiener process in E. Assuming the existence of an invariant measure μ for L, under suitable assumptions on A we show that the spectrum of L in the space Lp (E, μ) (1< p< ∞) is given by [Formula: see text] where Aμ is the generator of a Hilbert space contraction semigroup canonically associated with A and μ. We prove that the assumptions on A are always satisfied in the strong Feller case and in the finite-dimensional case. In the latter case we recover the recent Metafune–Pallara–Priola formula for σ(L).