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Properties of vector-valued submodules on the bidisk

In previous work, the authors studied the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\mathbb{D}^2)\ominus θH^2(\mathbb{D}^2)$, where $θ$ is a two-variable scalar inner function. Among other results, the authors used Agler decompositions to characterize the ranks of the operators $[S_{z_j}, S^*_{z_j}]$ in terms of the degree of rational $θ.$ In this paper, we examine similar questions for $H^2(\mathbb{D}^2)\ominus ΘH^2(\mathbb{D}^2)$ when $Θ$ is a matrix-valued inner function. We extend several results our previous work connecting $\text{Rank} [S_{z_j}, S^*_{z_j}]$ and the degree of $Θ$ to the matrix setting. When results do not clearly generalize, we conjecture what is true and provide supporting examples.