9501304 Rovnyak An approach to Krein space operator theory will be developed based on factorization and extension properties of operators. Continuous Krein space operators have natural isometric and unitary extensions, allowing a reduction of the study of general operators to a more tractable case. New tools in the form of index formulas give conditions for the existence of factorizations. A new form of the Krein space commutant lifting theorem is sought which has applications similar to the Hilbert space case. A related prmblem is to develop a unified algebraic and analytic theory of the Krein space Schur class of operator-valued holomorphic functions. This part is in collaboration with several European colleagues and features colligations to study indefinite kernels and reproducing kernel Krein spaces. Examples from interpolation theory and univalent functions will be considered. Factorization properties of selfadjoint operators on Krein spaces will be applied to the study of hyperinvariant subspaces. Other areas of application include compact perturbations of selfadjoint operators and a search for a Weyl-von Neumann theorem in Krein spaces. Numerical techniques will be used in combination with recent decompositions for finite matrices in an effort to obtain information on the general form of numerical range for finite-dimensional spaces. Interpolation problems and linear systems have long been important in pure and applied mathematics, as well as in technology. It is only recently that mathematicians have been looking into the possibility of generalizations to new settings such as indefinite inner product spaces. Preliminary results indicate that this line of investigation could turn out to be as fruitful as the classical theory has been. The problems naturally raise new questions about other areas of operator theory on indefinite inner product spaces. These have to do with structural properties of operators such a s invariant subspaces and numerical ranges. Connections with related areas of mathematics such as coefficient problems in classical complex analysis will also be explored. ***
