Pairs of inner projections and two applications

Orthogonal projections onto closed subspaces of $H^2({\mathbb{D}}^n)$ of the form $\phi H^2({\mathbb{D}}^n)$ for inner functions $\phi$ on ${\mathbb{D}}^n$ are referred to as inner projections, where $H^2({\mathbb{D}}^n)$ denotes the Hardy space over the open unit polydisc ${\mathbb{D}}^n$. In this paper, we classify pairs of commuting inner projections. We also present two seemingly independent applications: the first is an answer to a question posed by R. G. Douglas, and the second is a complete classification of partially isometric truncated Toeplitz operators with inner symbols on the polydisc.