On modules of the Hardy space of the Hartogs triangle
Abstract
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 double 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.
