Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces

Abstract

Motivated by the recent developments of de Branges-Rovnyak spaces, we investigate the function theoretic aspects of finite rank de Branges-Rovnyak spaces $H(B)$ generated by row-valued Schur functions $B$. We provide a generalization of Sarason's fundamental work by characterizing finite rank $H(B)$-spaces as the domain of the adjoint of the Toeplitz operators $T_{\phi }^{\ast }$ with symbol $\phi =B{A}^{-1}$, where $A$ is an matrix-valued outer function satisfying $A^{\ast }A+B^{\ast }B=I$ a.e. on the unit circle. We derive a norm formula for functions in $H(B)$-space and provide a concrete realization of this norm in terms of the Taylor coefficients of the function and the symbol $\phi$. As an application, we characterize all symbols $B$ for which $H^{\infty }\subseteq H(B)$ in terms of the boundary behavior of $I-B{B}^{\ast }$, thereby extending Sarason's criterion for the classical de Branges-Rovnyak spaces.

14 pages. Any comments and suggestions are welcome