$\delta$-function of an operator: A white noise approach

Let $(E) \subset (L^2) \subset (E)^*$ be the canonical framework of white noise analysis over the Gel'fand triple $S({\mathbb R}) \subset L^2({\mathbb R}) \subset S^*({\mathbb R})$ and ${\mathcal L} \equiv {\mathcal L}[(E),(E)^*]$ be the space of continuous linear operators from $(E)$ to $(E)^*$. Let $Q$ be a self-adjoint operator in $(L^2)$with spectral representation $Q = \int_{\mathbb R}\lambda\,P_Q(d\lambda)$. In this paper, it is proved that under appropriate conditions upon $Q$, there exists a unique linear mapping $Z:S^*({\mathbb R}) \longmapsto {\mathcal L}$ such that $Z(f)=\int_{\mathbb R}f(\lambda)\,P_Q(d\lambda)$ for each $f \in S({\mathbb R})$. The mapping is then naturally used to define $\delta(Q)$ as $Z(\delta)$, where $\delta$ is the Dirac $\delta$-function. Finally, properties of the mapping $Z$ are investigated and several results are obtained.