An analytic characterization of symbols of operators on white noise functionals
Let \(T\) be a topological space with Borel measure \(\nu\) and take a Gelfand triple \(E\subset L^ 2(T,\nu) \subset E^*\) constructed in a particular way. Then, by second quantization another Gelfand triple \((E)\subset L^ 2(E^*,\mu)\subset (E)^*\) is constructed, where \((E)\) and \((E)^*\) are the spaces of test and generalized white noise functionals on the Gaussian space \((E^*,\mu)\). In this paper general operators \(\Xi\in{\mathcal L}((E),(E)^*)\) are discussed. The symbol of \(\Xi\) is defined by \(\widehat {\Xi}(\xi,\eta)=\langle\langle \Xi \varphi_ \xi,\varphi_ \eta \rangle\rangle\), where \(\varphi_ \xi\in(E)\) is an exponential vector indexed by \(\xi\in E\). The main result is a characterization of such operator symbols in terms of analyticity and certain boundedness condition. A simple consequence of this characterization theorem is that every operator \(\Xi\in {\mathcal L}((E),(E)^*)\) admits an infinite series expansion in terms of integral kernel operators with norm estimates (Fock expansion). Here an integral kernel operator is by definition a superposition of creation and annihilation operators with distribution \(\kappa\) as integral kernel: \[ \Xi(\kappa)= \int_{T^{\ell+m}} \kappa(s_ 1,\dots, s_ \ell, t_ 1,\dots, t_ m) \partial_{s_ 1} \dots \partial_{s_ \ell} \partial_{t_ 1}^*\dots \partial_{t_ m}^* ds_ 1\dots ds_ \ell dt_ 1\dots dt_ m, \] where \(\partial_ t\) and \(\partial_ t^*\) are annihilation and creation operators at a point \(t\in T\), respectively. It is known that \(\partial_ t\in {\mathcal L}((E),(E))\), \(\partial_ t^*\in {\mathcal L}((E)^*, (E)^*)\) and \(\Xi(\kappa)\in {\mathcal L}((E),(E)^*)\) in general. Examples of Fock expansion include Laplacians, Kuo's Fourier transform, Weyl form of CCR, integral-sum kernel operators in quantum probability and so on.
superposition of creation and annihilation operators
integral-sum kernel operators in quantum probability
scientific article; zbMATH DE number 469070