On modules of the Hardy space of Hartogs triangle

Abstract

In this paper, we investigate the structure of doubly commuting submodules and quotient modules of the Hardy space $H^2({△}_H)$ 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 $({\theta }_1(z/w){\theta }_2(w)H^2({△}_H){)}^{\perp }$, where ${\theta }_1$ and ${\theta }_2$ are inner functions on the unit disc. This is achieved by introducing the concept of $\phi$-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 $(p{H}^2({△}_H){)}^{\perp }$ under some mild assumptions on $p$, where $p$ is a polynomial in two variables.

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