Generalized Bessel Operator and Reproducing Kernel Theory

In 1961, Bargmann introduced the classical Fock space FC and in 1984, Cholewinsky introduced the generalized Fock space F2,νC. These two spaces are the aim of many works, and have many applications in mathematics, in physics and in quantum mechanics. In this work, we introduce and study the Fock space F3,νC associated to the generalized Bessel operator L3,ν. The space F3,νC is a reproducing kernel Hilbert space (RKHS). This is the reason for defining the orthogonal projection operator, the Toeplitz operators and the Hankel operators associated to this space. Furthermore, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator T:F3,νC→H, where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is a difference operator and an integral operator, respectively. Finally, we remark that it is now natural to raise the problem of studying the Bessel-type Segal-Bargmann transform associated to the space F3,νC. This problem is difficult and will be an open topic. This topic requires more details for the harmonic analysis associated to the operator L3,ν. We have the idea to continue this research in a future paper.