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

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