Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

Abstract: In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on $\mathfrak{X}$, where $\mathfrak{X}$ is a complex seperable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS $\mathcal{B}(\mathfrak{E})$ based on the de Branges operator $\mathfrak{E}=(E_-,E_+)$ has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been given.