On the one-mode quadratic Weyl operators

It has been known, in [7], that one-mode quadratic Weyl operators are well-defined as unitary operators acting on the quadratic Fock space. These operators were defined by their action on the finite particles space. However, their action on the domain of the quadratic exponential vectors is still unknown. In this paper, we provide a new reformulation (w.r.t. [1]) of the quadratic exponential vectors and we compute the action of the one-mode quadratic Weyl operators on the set of these exponential vectors. Then, we prove the independence of the one-mode quadratic Weyl operators parameterized by the principal domain associated with the quadratic Heisenberg group obtained in [7]. This significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the C⁎C⁎-representation of the renormalized square of white noise algebra.