De Branges-Rovnyak spaces generated by row Schur functions with mate

TL;DR

This paper investigates de Branges-Rovnyak spaces generated by row Schur functions with mates, establishing polynomial density, characterizing invariant subspaces, and describing cyclic vectors under finite rank conditions.

Contribution

It provides new characterizations of subspaces and cyclic vectors in de Branges-Rovnyak spaces generated by row Schur functions with mates.

Findings

Polynomials are dense in $\\mathcal{H}(B)$.

Backward shift invariant subspaces are characterized.

Cyclic vectors are described for finite rank cases.

Abstract

In this paper, we study the de Branges-Rovnyak spaces $\mathcal{H}(B)$ generated by row Schur functions $B$ with mate $a$. We prove that the polynomials are dense in $\mathcal{H}(B)$, and characterize the backward shift invariant subspaces of $\mathcal{H}(B)$. We then describe the cyclic vectors in $\mathcal{H}(B)$ when $B$ is of finite rank and $\dim (aH^2)^\perp < \infty$.