On modules of the Hardy space of Hartogs triangle

In this paper, we investigate the structure of doubly commuting submodules and quotient modules of the Hardy space $H^2(\triangle_H)$ over the Hartogs triangle. We establish a complete classification of doubly commuting submodules. In addition, we characterize all doubly commuting quotient modules of the form $(\theta_1(z/w)\theta_2(w)H^2(\triangle_H))^\perp$, where $\theta_1$ and $\theta_2$ are inner functions on the unit disc. This is achieved by introducing the concept of $\varphi$-doubly commuting quotient modules on the Hardy space $H^2(\mathbb D^2).$ We further explore the essential normality and doubly commutativity of quotient modules of the form $(pH^2(\triangle_H))^\perp$ under some mild assumptions on $p$, where $p$ is a polynomial in two variables.