Structured Invariant Spaces of Vector Valued Rational Functions, Hermitian Matrices, and a Generalization of the lohvidov Laws

Finite dimensional indefinite inner product spaces of vector valued rational functions which are (1) invariant under the generalized backward shift and (2) subject to a structural identity, and subspaces and “superspaces” thereof are studied. The theory of these spaces is then applied to deduce a generalization of a pair of rules due to Iohvidov for evaluating the inertia of certain subblocks of Hermitian Toeplitz and Hermitian Hankel matrices. The connecting link rests on the identification of a Hermitian matrix as the Gram matrix of a space of vector valued functions of the type considered in the first part of the paper. Corresponding generalizations of another pair of theorems by Iohvidov on the rank of certain subblocks of non-Hermitian Toeplitz and non-Hermitian Hankel matrices are also stated, but the proofs will be presented elsewhere. *Current address: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. ‘H. Dym would like to thank Renee and Jay Weiss for endowing the chair which supports his research. LINEAR ALGEBRA AND ITS APPLlCATlONS 137/138:137-181(1990) 0 Elsevier Science Publishing Co., Inc., 1990 137 655 Avenue of the Americas, New York, NY 10010 0024-3795/90/$3.50 138 DANIEL ALPAY AND HARRY DYM