Vector valued de branges spaces of entire functions and related topics

Abstract

This thesis primarily deals with vector valued reproducing kernel Hilbert spaces (RKHS) H of entire functions associated with operator valued kernel functions. de Branges operators E = (E−,E+) are introduced as a pair of Fredholm operator valued entire functions on X, where X is an infinite dimensional complex separable Hilbert space. A few explicit examples of these de Branges operators are discussed. We highlight that the newly defined RKHS B(E) based on the de Branges operator E = (E−,E+) generalizes Paley-Wiener spaces of vector valued entire functions. These spaces are 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 space B(E) has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with, and it has been shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. A brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges space B(E) has been given. We discuss an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within an RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been studied. Furthermore, we establish the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions. Some factorization and isometric embedding results are extended from the scalar valued theory of de Branges spaces. In particular, global factorization of Fredholm operator valued entire functions and analytic equivalence of reproducing kernels of de Branges spaces are discussed. Additionally, the operator valued entire functions associated with these de Branges spaces are studied, and a connection with operator nodes is established. We extend the concept of de Branges matrices to any finite m×m order where m = 2n. We shall discuss these matrices along with the theory of de Branges spaces of Cn-valued entire functions and their associated functions. A parametrization of these matrices is obtained using the Smirnov maximum principle for matrix valued functions. Additionally, a factorization of matrix valued meromorphic functions is discussed.