# Invariant subspaces of $\mathcal{H}^2(\mathbb{T}^2)$ and $L^2(\mathbb{T}^2)$ preserving compatibility

## TL;DR

This paper characterizes invariant subspaces of the Hardy space and L^2 space over the torus where multiplication operators by independent variables are compatible, advancing understanding of operator invariance in these function spaces.

## Contribution

It provides a detailed description of invariant subspaces with compatible multiplication operators on Hardy and L^2 spaces over the torus, a novel analysis in this context.

## Key findings

- Characterization of invariant subspaces with compatible operators
- Description of invariant subspaces in Hardy and L^2 spaces
- Advancement in understanding operator invariance on the torus

## Abstract

Operators of multiplication by independent variables on the space of square summable functions over the torus and its Hardy subspace are considered. Invariant subspaces where the operators are compatible are described.