Hilbert spaces contractively included in the Hardy space of the bidisk

We study the reproducing kernel Hilbert spaces Heng hooktop sign (double-struck D², S) with kernels of the form I - S(z₁, Z₂ >)S(w₁, w₂)*/ (1 - z₁ w^*₁)(1 - Z₂w^*₂) where S(z₁ ,z₂) is a Schur function of two variables z₁ ,z₂ ∈ double-struck D. They are analogs of the spaces Heng hooktop sign(double-struck D, S) with reproducing kernel (1 -S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak in L. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of Heng hooktop sign(Double-struck D₂, S) as a subspace of the Hardy space on the bidisk. The spaces Heng hooktop sign(double-struck D₂, S) form a proper subset of the class of the so-called sub-Hardy Hilbert spaces of the bidisk.