Abstract interpolation in vector-valued de Branges-Rovnyak spaces

TL;DR

This paper investigates a general metric constrained interpolation problem within vector-valued de Branges-Rovnyak spaces, extending classical results and exploring connections with Toeplitz operators and shift-invariant subspaces.

Contribution

It provides a comprehensive description of all solutions to the interpolation problem in de Branges-Rovnyak spaces, including special cases and injectivity conditions.

Findings

Characterization of all solutions via bounded functions in de Branges-Rovnyak spaces

Extension of vector-valued interpolation problems with operator arguments

Connections established with Toeplitz kernels and backward shift subspaces

Abstract

Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.