Beurling-type invariant subspaces of the Poletsky–Stessin–Hardy spaces in the bidisc

Abstract

The invariant subspaces of the Hardy space \(H^2(\mathbb {D})\) 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 structure of invariant subspaces of Poletsky–Stessin–Hardy spaces which are the 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 to the vector-valued Poletsky–Stessin–Hardy spaces and then we gave a necessary and sufficient condition for the invariant subspaces of \(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) to be of Beurling-type.