3 problems

Let $\Theta$ be the two-variable inner function under consideration, and let $S^1_{\Theta}$ denote the corresponding compression of the shift. Write $\theta$ and $\theta_1$ for the…

Let $\Theta$ be a $d\times d$ matrix-valued inner function on the bidisk, and let $S_{z_1}$ denote the associated compressed shift operator. Write $\deg_1\Theta$ and…

Unbounded-face conjecture. Every $\gamma=(g_1,g_2)$ where both $g_i$'s are hyperbolic admits a Dirichlet domain with more than two faces.