Reproducing Kernel Hilbert Spaces, Matrix Theory, their relations and applications

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This proposal consists of several interrelated tasks on fundamental problems in modern complex and functional analysis, matrix analysis and their applications to other fields of mathematics and engineering, e.g., approximation theory, mathematical physics, control theory, signal processing and electrical engineering. Analytic Function Spaces and the operators acting on them has been an active domain of research. RKHS provide a modern and powerful tool to look at such problems, classic and new, and thus they play an important role in numerous domains of applied and pure sciences. The advent of reproducing kernels goes back to the founding works of several prominent mathematicians like Nevanlinna, Pick, and Schur on exact constrained interpolation. Since then, RKHS made evidence of their central role and strength in the study of properties of a wide range of spaces as demonstrated by breakthrough results on interpolation, sampling, uniqueness, and invariant subspaces by Aleman, Carleson, Fricain, Ransford, Richter, Sarason, Seip, etc. The solution in 2013 of the Feichtinger conjecture is a milestone and opens new research directions in the field of reproducing kernels.We consider several such spaces, e.g., Hardy, Dirichlet, Bergman, Model and de Branges-Rovnyak spaces. The most celebrated operators on these spaces are the forward and backward shift operators. These objects lead to more general concepts like Toeplitz, Hankel operators, Berezin transform and composition operators. Any such operator can be interpreted as an infinite dimensional matrix acting on the sequence space formed with the coefficients of functions in the ambient space. To treat infinite dimensional matrices, we naturally consider their truncations and thus the classical matrix theory shows its face. Hence, looking from this angle, techniques of matrix theory (infinite dimensional as well as finite dimensional) are applied in RKHS. Geometric properties of families of reproducing kernels like completeness, minimality, being a Riesz basis or an asymptotically orthonormal basis, are intimately related to properties like interpolation, sampling, and uniqueness in spaces of holomorphic functions. We are mainly interested here in Hardy, Dirichlet and model spaces and their generalization de Branges-Rovnyak spaces. This leads us to study uniqueness sets and zero sets, cyclicity, and interpolating and sampling sequences in Dirichlet and de Branges-Rovnyak spaces as well as in model subspaces of Hardy spaces. They have natural applications in spectral theory, generalized Hardy spaces, norm control of matrix inversion, and control theory. Moreover, we encounter questions which are interesting in their own right in the subject of matrix theory. A celebrated question, which is the continuation of an old conjecture, is the loci of eigenvalues of doubly-stochastic matrices.