Structured Invariant Spaces of Vector Valued Functions, Hermitian Forms and a Generalization of Iohvidov's Laws
Vector spaces of pairs of rational vector valued functions, which are (1) invariant under the generalized backward shift and (2) endowed with a sesquilinear form which is subject to a structural identity, are studied. It is shown that any matrix can be viewed as the “Gram” matrix of a suitably defined basis for such a space. This identification is used to show that a rule due to Iohvidov for evaluating the rank of certain subblocks of a Toeplitz (or Hankel) matrix is applicable to a wider class of matrices with (appropriately defined) displacement rank equal to two. Enroute, a theory of reproducing kernel spaces is developed for nondegenerate spaces of the type mentioned above.
D. Alpay and H. Dym. Structured invariant spaces of vector valued functions, hermitian forms and a generalization of Iohvidov's laws. In Linear Algebra and its Applications, vol. 137 (1990), pp. 137-181.