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**1 - 2**of**2**### SECOND QUANTIZATION AND THE L p-SPECTRUM OF NONSYMMETRIC ORNSTEIN-UHLENBECK OPERATORS

, 2005

"... 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-dimens ..."

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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 dU(t) = AU(t) dt + dW(t) 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 L p (E, µ) (1 < p < ∞) is given by { ∑n σ(L) = j=1 kjzj: kj ∈ N, zj ∈ σ(Aµ); j = 1,..., n; n ≥ 1, 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). 1.

### © Hindawi Publishing Corp. ON GROMOV’S THEOREM AND L 2-HODGE DECOMPOSITION

, 2002

"... Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the corresponding L 2-harmonic sections. In particular, some known results concerning Gro ..."

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Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the corresponding L 2-harmonic sections. In particular, some known results concerning Gromov’s theorem and the L 2-Hodge decomposition are considerably improved. 2000 Mathematics Subject Classification: 58J50, 58J65. 1. Introduction. Recall