The core function of submodules over the bidisk

The Hardy space H 2 (D 2 ) over the bidisk is a module over the polynomial ring C[z 1 , z 2 ]. This paper studies two natural objects associated with submodules in H 2 (D 2 ), namely the core fraction and the core operator. Boundary values of the core function exhibit. a Beurling-type phenomenon, and the Hilbert-Schmidt norm of the core operator is shown to be an important numerical invariant of M. For a homogeneous ideal I ⊂ C[z i ,z 2 ], let I = pL be its Beurling form; then it is shown that the core operator for the submodule [I] is of finite rank if and only if p = z k 1z 2 for some nonnegative integers k, .