Abstract:In this article, using Halmos' two projections theorem, we completely characterize the essential commutativity of the orthogonal projections onto the shift-invariant subspaces $\phi_1 H^2(\mathbb{D})$ and $\phi_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ via local conditions on the inner functions $\phi_1$ and $\phi_2$. Finite-rank commutators $[P_{\phi_1}, P_{\phi_2}]$ are also characterized. Using our methods, we connect the essential commutativity with the Fredholmness of the projections $(P_{\phi_1}, P_{\phi_2})$ 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.
From: Srijan Sarkar [view email]
[v1]
Wed, 24 Jun 2026 10:07:33 UTC (27 KB)
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