Exponential of the Quadratic Creation Operator

This paper deals with the Fock representation of the re–normalized square of the white noise algebra (RSWN), realized through the quadratic analogue of the standard Fock space known as the quadratic Fock space. Similarly to the usual Fock space, the quadratic Fock space \(\varGamma (\mathcal {K})\) $\Gamma (\mathcal{K})$ is equipped with three fundamental operators: creation, annihilation, and preservation. However, in contrast to the standard case, a thorough analytic study of these operators is lacking. The primary focus of this paper is on the study of the creation operator \(B^+_f\) $B_f^{+}$ . Specifically, we establish that the set of quadratic exponential vectors corresponding to test functions with \(\infty \) $\infty$ –norm less than a real number r within the interval (0, 1] is total in \(\varGamma (\mathcal {K})\) $\Gamma (\mathcal{K})$ . Additionally, we prove that, for a specific choice of r, this set is contained in the domain of \(e^{B^+_f}\) $e^{B_f^{+}}$ . Lastly, we explicitly describe the action of \(e^{B^+_f}\) $e^{B_f^{+}}$ on the mentioned set of quadratic exponential vectors.