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 lohvidov 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 lohvidov on the rank of certain subblocks of non-Hermitian Teoplitz and non-Hermitian Hankel matrices are also stated, but the proofs will be presented elsewhere.