The research in this proposal is concerned with the theory of matrices and operators. In operator theory the concern will be in operator valued functions and the central theme will be factorization problems. Some of the topics are analysis of spectral data of various classes of operator functions, synthesis of an operator function in a given class given its spectral data, special factorizations of self adjoint operator polynomials, correspondence between factorizations of an analytic operator function with compact spectrum and invariant subspaces of its linearization and local and global equivalence of meromorphic operator functions. In matrix theory the problems will include matrix completion problems, spectral distribution in qualitative matrix theory, infinite dimensional spectral assignment problems, linear transformations that preserve controllability, a generalization of the characteristic polynomial, and analytic families of subspaces. A general goal will be the understanding of the unifying structure which may support additional problem solutions in each area.
