  Orthogonal projections onto closed subspaces of $H^2(\mathbb{D}^n)$ of the
form $\varphi H^2(\mathbb{D}^n)$ for inner functions $\varphi$ 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.

Functional Analysis;Complex Variables;Operator Algebras