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.
Categories: math.FA math.CV math.OA