Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Let H2(γ) be the Hilbert space over the bidisk \(\mathbb{D}^2\) generated by a positive sequence γ = {γnm}n,m≥0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H2(γ) with γ = {γnm}n,m≥0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk \(\mathbb{D}^2\).