momentum operators to which the original problem is mapped by appropriated integral transform such as the Bargmann transform.
variable, thus its link with quantum mechanics and the Hamiltonian operator. Similarly (and again up to a positive multiplicative factor) the Hardy space can be seen as the only space of functions analytic in the open unit disk for which the adjoint of the backward shift operator is the multiplication operator.
Calogero-Sutherland-Moser models or fractional derivatives like in the cased of grey noise stochastic processes based on the Mittag-Leffler function as probability measure. These type of derivatives can be considered as special cases of the Gelfond-Leontiev operator of generalized differentiation.
In this paper we present a general framework for construction and study of Fock and Hardy spaces with respect to the Gelfond-Leontiev operator of generalized differentiation.
We propose a characterization of the Hardy space in term of the adjoint of such generalized fractional differentiation operator. We begin by an appropriated definition of such Hardy spaces using reproducing kernel methods. This leads to a Carleman's condition associated to the correspondent Stieltjes moment problem and will allow for a new characterization of the Fock space.