Hilbert spaces contractively included in the Hardy space of the bidisk

Abstract

We study the reproducing kernel Hilbert spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) with kernels of the form $$\frac{{I - S(z_1 ,z_2 > )S(w_1 ,w_2 )^* }}{{(1 - z_1 w_1^* )(1 - z_2 w_2^* )}}$$ where S(z1,z2) is a Schur function of two variables z1,z2l\(\mathbb{D}\). They are analogs of the spaces \(\mathfrak{H}\left( {\mathbb{D},S} \right)\) with reproducing kernel (1-S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak l. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) as a subspace of the Hardy space on the bidisk. The spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) form a proper subset of the class of the so–called sub–Hardy Hilbert spaces of the bidisk.