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.

单位圆盘的哈代空间\(H^2(\mathbb {D})\)的不变子空间是众所周知的；然而，在几个变量中，经典哈代空间的不变子空间的结构尚未完全理解。在这项研究中，我们研究了 Poletsky-Stessin-Hardy 空间的不变子空间的结构，这些空间是经典 Hardy 空间到\(\mathbb {C}^n\)中超凸域的推广。我们证明了不是\(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) 的所有不变子空间属于 Beurling 型。为了刻画这个空间的 Beurling 型不变子空间，我们首先将 Lax-Halmos 定理推广到向量值 Poletsky-Stessin-Hardy 空间，然后我们给出了\(H^{ 2}_{\tilde{u}}(\mathbb {D}^2)\)是 Beurling 类型的。