Pairs of inner projections and two applications

TL;DR

This paper classifies pairs of commuting inner projections on Hardy spaces over the polydisc and applies the results to answer a question by Douglas and classify certain truncated Toeplitz operators.

Contribution

It provides a classification of pairs of commuting inner projections and applies this to solve an open problem and classify specific truncated Toeplitz operators.

Findings

Classification of pairs of commuting inner projections.

Answer to R. G. Douglas's question.

Complete classification of certain truncated Toeplitz operators.

Abstract

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.