The purpose of this thesis is to study certain reproducing kernel Krein spaces of analytic functions, the relationships between these spaces and an inverse scattering problem associated with matrix valued functions of bounded type, and an operator model.
Roughly speaking, these results correspond to a generalization of earlier investigations on the applications of de Branges' theory of reproducing kernel Hilbert spaces of analytic functions to the inverse scattering problem for a matrix valued function of the Schur class.
The present work considers first a generalization of a portion of de Branges' theory to Krein spaces. We then formulate a general inverse scattering problem which includes as a special case the more classical inverse scattering problem of finding linear fractional representations of a given matrix valued function of the Schur class and use the theory alluded to above to obtain solutions to this problem.
Finally, we give a model for certain hermitian operators in Pontryagin spaces in terms of multiplication by the complex variable in a reproducing kernel Pontryagin space of analytic functions.
D. Alpay, "Reproducing Kernel Krein Spaces of Analytic Functions and Inverse Scattering," Ph.D. dissertation, Weizmann Institute of Science, Rehovot, Israel, 1985.