Analytic Characterization for HilbertSchmidt Operators on Fock Space

In this paper we first introduce and investigate some special classes of nonlinear maps and get several useful results. Then, by using these results, we obtain analytic criteria for checking whether or not an operator defined only on the exponential vectors of a symmetric Fock space becomes a HilbertSchmidt operator on the whole space. Additionally, as an application, we also get an analytic criterion for HilbertSchmidt operators on a Gaussian probability space through the WienerItSegal isomorphism.