Integral representation of second quantization and its application to white noise analysis
Let \(K\) denote a continuous linear operator on the underlying basic space \(E\) in white noise analysis. It is shown that the second quantization \(\Gamma(K)\) has the integral representation \[ \Gamma(K) \varphi(x)= \int_{E'} \int_{E'} \widetilde{\varphi} (K^*x+ iK^*y+z)\mu(dy)\mu(dz), \tag{1} \] where \(\widetilde{\varphi}\) is the analytic version of a white noise (or Hida's) test function \(\varphi\in(E)\) and \(\mu\) is the white noise measure. If there exists a continuous linear operator \(L\) on \(E\) such that \(I-K^*K= L^*L\), then (1) can be rewritten into a simpler form \[ \Gamma(K) \varphi(x)= \int_{E'} \widetilde{\varphi} (K^*x+ L^*y) \mu(dy). \tag{2} \] Eq. (2) is also called Mehler's formula especially when \(K\) is a contraction on the underlying Hilbert space \(H\) and \(L=\sqrt{I-K^*K}\). It is interesting to note that, in a particular case, (1) is replaced by a \(L^2(E', d\mu)\)-limit form of the similar integral expression, which is close connected with the Ornstein-Uhlenbeck semigroup introduced by \textit{I. Shigekawa} [J. Math. Kyoto Univ. 32, No. 4, 731-748 (1992; Zbl 0777.60047)]. Let \(A\) be a densely defined positive selfadjoint operator on \(H\), called a standard operator in the framework of white noise calculus. Applications of such a representation to the case \(K=A^p\) allow some sharper estimates on the norm \(|\varphi|_{2,p}= |\Gamma (A^p)\varphi|_2\). Moreover, the author establishes a topological equivalence between the Meyer-Yan space \({\mathcal M}\) and the Lie space \({\mathcal E}\) [cf. \textit{Y.-J. Lee}, J. Funct. Anal. 82, No. 2, 429-464 (1989; Zbl 0683.46035)].
densely defined positive selfadjoint operator
scientific article; zbMATH DE number 850609