Essentially commuting projections onto shift-invariant subspaces

Abstract

In this article, using Halmos' two projections theorem, we completely characterize the essential commutativity of the orthogonal projections onto the shift-invariant subspaces $ϕ_1 H^2(\mathbb{D})$ and $ϕ_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ via local conditions on the inner functions $ϕ_1$ and $ϕ_2$. Finite-rank commutators $[P_{ϕ_1}, P_{ϕ_2}]$ are also characterized. Using our methods, we connect the essential commutativity with the Fredholmness of the projections $(P_{ϕ_1}, P_{ϕ_2})$ as introduced by Avron, Seiler and Simon. Applications include refining existing conditions for compactness of truncated Toeplitz operators corresponding to inner symbols and thereby characterizing the compactness of certain contractions using the Sz.-Nagy--Foias model theory. We conclude with several characterizations on the polydisc.