A role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space
The authors start out with a rigged Fock space, defined in a natural way in terms of a self-adjoint operator with spectrum in \([1, \infty)\). The technical point here is that this rigging need not be nuclear. Given this setup, the symbol of a given continuous linear operator is defined in terms of coherent (or exponential) vectors. Then necessary and sufficient conditions are given that a function be the symbol of a continuous operator or of a Hilbert-Schmidt operator (with respect to various possibilities for the domain and codomain of the operator); see Theorem 5.2, the main theorem, as well as Theorems 5.3, 5.4, 5.5 and 6.2. Two applications of the main theorem are presented. First, in Theorem 7.3 and Corollary 7.4, a Wiener-Itô (or chaotic) expansion is given for a continuous operator (with respect to the same choices for the domain and codomain), while in Theorem 7.6 this is done for the Hilbert-Schmidt class. In Section 8, this Wiener-Itô expansion is related to the Fock expansion. Finally, the operator-valued solution curve of \[ \frac{d {\Xi} }{dt} = L_t \diamond { \Xi}, \quad{\Xi}(0) = I, \] a white noise differential equation, is proved to lie in an appropriate space of continuous linear operators if the quantum stochastic process \(L_t\) satisfies certain hypotheses. Here, \(\diamond\) denotes the Wick (or normal-ordered) product.
scientific article; zbMATH DE number 2110884