Rigged de Branges–Pontryagin spaces and their application to extensions and embedding

Abstract

De Branges–Pontryagin spaces B(E) with negative index κ of entire p×1 vector valued functions based on an entire p×2p entire matrix valued function E(λ) (called the de Branges matrix) are studied. An explicit description of these spaces and an explicit formula for the indefinite inner product are presented. A characterization of those spaces B(E) that are invariant under the generalized backward shift operator that extends known results when κ=0 is given. The theory of rigged de Branges–Pontryagin spaces is developed and then applied to obtain an embedding of de Branges matrices with negative squares in generalized J-inner matrices and selfadjoint extensions of the multiplication operator in B(E). A formula for factoring an arbitrary generalized J-inner matrix valued function into the product of a singular factor and a perfect factor is found analogous to the known factorization formulas for J-inner matrix valued functions.