ON A NEW CLASS OF REPRODUCING KERNEL SPACES AND A NEW GENERALIZATION OF THE IOHVIDOV LAWS

A new class of finite dimensional reproducing kernel spaces of m X 1 vector valued analytic functions on a fairly general domain OMEGA+ is introduced. The reproducing kernels of these spaces have a special form which is based on an m X m matrix valued function THETA, which is J unitary on the boundary of OMEGA+. Every invertible Hermitian matrix can be interpreted as the Gram matrix of a suitably chosen basis in such a space, where m is equal to the appropriately defined displacement rank of the given matrix. Orthogonal decompositions of the space are developed in terms of Schur complements of the matrix and factorizations of the THETA, much as in the classical cases wherein OMEGA+ is either the unit disc or the half plane. Finally, a new generalization of the Iohvidov laws (which extends earlier generalizations by the same authors) is deduced as an application of the theory.