Beurling-Type Invariant Subspaces of the Poletsky-Stessin Hardy Spaces in the Bidisc

Abstract

The invariant subspaces of the Hardy space on $H^2(\mathbb{D})$ of the unit disc are very well known however in several variables the structure of the invariant subspaces of the classical Hardy spaces is not yet fully understood. In this study we examine the invariant subspace problem for Poletsky-Stessin Hardy spaces which is a natural generalization of the classical Hardy spaces to hyperconvex domains in $\mathbb{C}^n$. We showed that not all invariant subspaces of $H^{2}_{\tilde{u}}(\mathbb{D}^2)$ are of Beurling-type. To characterize the Beurling-type invariant subspaces of this space we first generalized the Lax-Halmos theorem of vector valued Hardy spaces to the vector valued Poletsky-Stessin Hardy spaces and then we give a necessary and sufficient condition for the invariant subspaces of $H^{2}_{\tilde{u}}(\mathbb{D}^2)$ to be of Beurling-type.