Executive Summary : | The study of reproducing kernel Hilbert spaces (RKHS) grew out of work on integral operators by J. Mercer in 1909 and S. Bergman's work in complex analysis on various domains. Applications of RKHS can be found in many areas such as complex analysis, group representation theory, metric embedding theory, statistics, probability, the study of integral operators etc. The RKHS H(U) was introduced by L. de Branges, who developed the theory of this space and applied it to a number of problems in analysis, including inverse problems for canonical differential systems for m = 2. de Branges' early papers focused on entire J-inner matrix valued functions (mvf's). Later many mathematicians contributed to this theory and a lot of generalizations and representations have come up. In 1971 Harry Dym showed that de Branges spaces can be constructed from the unique solutions of certain system of differential equations with initial conditions. Also vector valued de Branges spaces associated with matrix valued kernels were studied and several applications of these spaces were established . The de Branges-Rovnyak spaces H(b) introduced by de Brange and Rovnyak are natural reproducing kernel spaces, generalizing the model spaces that appear prominently in the theory of contractions. Indefinite inner product space is a real or complex vector space with a hermitian form that is linear in first coordinate and conjugate linear with respect to second coordinate. Indefinite inner product spaces first appeared in a paper of Dirac on quantum field theory. Krein space and Pontryagin spaces are its special cases. The study of J-symmetric operators were introduced by Glazman which has now become a well established area of research in the areas related to linear differential operators with complex valued coefficient. Glazman observed that some non symmetric differential operators satisfies J-symmetricity condition in L2 and accordingly he defined the notion of J-selfadjoint operator. Then questions related to existence and characterization of J-selfadjoint extension of J-symmetric operators gained main focus of research. In this research proposal vector valued H(U), de Branges spaces are considered that have operator valued kernel functions. A question on de Branges-Rovnyak space is proposed. Also some analysis on reproducing kernel Krein spaces (RKKS) and reproducing kernel Pontryagin spaces (RKPS) are consided. Some problems on J-symmetric operators are proposed. |