Vector valued de Branges spaces, CNU contractions and functional models

TL;DR

This paper explores vector valued de Branges spaces linked to de Branges operators, demonstrating their role as functional models for certain contraction operators and connecting them with characteristic functions.

Contribution

It introduces a new class of vector valued de Branges spaces constructed via Hilbert space decompositions and relates them to operator model theory.

Findings

Constructed vector valued de Branges spaces using Hilbert space decompositions.

Showed these spaces serve as functional models for non-unitary contraction operators.

Established that the characteristic function equals the projection operator function on the unit disc.

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 these are vector valued de Branges spaces. We further demonstrate that these spaces provide functional models for certain completely non-unitary contraction operators. 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. These results provide a new perspective on the role of vector valued de Branges spaces in operator model theory.