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Functional calculus for the Ornstein-Uhlenbeck operator. (English) Zbl 0995.47010
Let $$\gamma$$ be the Gauss probability measure on $$R^d$$ with density $$\gamma_0 (x)=\pi^{-d/2} \exp(-|x |^2)$$. Then the Ornstein-Uhlenbeck operator $$-1/2 \Delta +x\cdot \nabla$$ is essentially selfadjoint in $$L^2 (\gamma)$$. Denote by $$L$$ its selfadjoint extension. It is well-known that the spectrum of $$L$$ is $$\{0, 1, \dots \}$$. Let $$\{ P_n\}$$ be the spectral resolution of the identity for which $$Lf=\sum_{n=0}^{\infty} nP_n f, f \in \text{Dom}(L)$$. The purpose of this paper is to develop a functional calculus for $$L$$, i.e., to find sufficient conditions on the spectral multiplier $$M=\{ M(n): n=0, 1, \dots, \}$$ for the spectral operator $$M(L)=\sum_{n=0}^{\infty} M(n)P_n$$, initially defined in $$L^2 (\gamma) \cap L^p (\gamma)$$, to extend to a bounded operator on $$L^p (\gamma)$$, for some $$p \in (1, \infty)$$.

##### MSC:
 47A60 Functional calculus for linear operators 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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