# A conservative de Branges-Rovnyak functional model for operator Schur functions on $\mathbb C^+$

## TL;DR

This paper develops a new conservative functional model for operator-valued Schur functions on the right-half plane, extending classical disk models to continuous-time systems with unbounded operators.

## Contribution

It introduces a de Branges-Rovnyak model directly in the right-half plane, incorporating unbounded connecting operators and non-invertible intertwinements for enhanced uniqueness.

## Key findings

- Model exhibits structure absent in disk setting
- Handles unbounded connecting operators in continuous-time systems
- Strengthens classical uniqueness results

## Abstract

We present a solution of the operator-valued Schur-function realization problem on the right-half plane by developing the corresponding de Branges-Rovnyak canonical conservative simple functional model. This model corresponds to the closely connected unitary model in the disk setting, but we work the theory out directly in the right-half plane, which allows us to exhibit structure which is absent in the disk case. A main feature of the study is that the connecting operator is unbounded, and so we need to make use of the theory of well-posed continuous-time systems. In order to strengthen the classical uniqueness result (which states uniqueness up to unitary similarity), we introduce non-invertible intertwinements of system nodes.