Cyclicity of reproducing kernels in weighted Hardy spaces over the bidisk

In general, the Beurling theorem does not hold for an invariant subspace in the Hardy space over the bidisk. In 1991, Nakazi posed a conjecture that the Beurling theorem holds for a singly generated invariant subspace. In this paper, a relation between a singly generated invariant subspace and a weighted Hardy space over the bidisk is studied. It is showed that there exists a weighted Hardy space over the bidisk which has a non-cyclic reproducing kernel. Also a counterexample for Nakazi's conjecture is given.