Vector valued de Branges spaces, CNU contractions and functional models

Abstract

In this paper, we study vector valued de Branges spaces associated with a de Branges operator, defined as a pair of Fredholm operator valued analytic functions on a domain symmetric with respect to the unit circle. Using a suitable direct sum decomposition of a Hilbert space, we construct a class of vector valued reproducing kernel Hilbert spaces and show that under some assumptions these are vector valued de Branges spaces. We further demonstrate that these spaces provide functional models for certain class of completely non-unitary contraction operators. We also give a Fredholm-type criterion for verifying the hypotheses of the main construction and apply it to several concrete classes of completely non-unitary contractions. Next, we establish connections between the Sz.-Nagy-Foias characteristic function of the contraction operator, the projection operator valued function arising from the Hilbert space decomposition, and the reproducing kernel of the de Branges space. In particular, we show that the characteristic function coincides with the projection operator valued function on the unit disc. Enroute, we also obtain a complete unitary invariance of a certain class of cnu contractions in terms of de Branges quotient operator valued functions. Finally, we discuss certain aspects of the canonical contraction in de Branges model and its $L^2$ realization. These results provide a new perspective on the role of vector valued de Branges spaces in operator model theory.