Finite-rank commutators of projections onto shift-invariant subspaces

[2606.25649] Finite-rank commutators of projections onto shift-invariant subspaces Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2606.25649 (math) [Submitted on 24 Jun 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Finite-rank commutators of projections onto shift-invariant subspaces Authors: Rounak Biswas , Srijan Sarkar View PDF HTML (experimental) Abstract: In this article, using Halmos' two projections theorem, we completely characterize finite-rank commutators of orthogonal projections onto shift-invariant subspaces $\phi_1 H^2(\mathbb{D})$ and $\phi_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ corresponding to inner functions $\phi_1, \phi_2$ on the unit disc $\mathbb{D}$. Using our methods, we connect the finite-rank commutator with the Fredholmness of the projections $(P_{\phi_1}, P_{\phi_2})$ as introduced by Avron, Seiler and Simon. We conclude with several characterizations on the polydisc. Comments: The results on compactness of commutators of projections in the earlier version contained errors, particularly Theorem 2.6 and its consequences. The article has been thoroughly revised, with the errors and corresponding references removed. The present version establishes correct results on finite-rank commutators of projections and compactness of products of projections Subjects: Functional Analysis (math.FA) ; Complex Variables (math.CV) MSC classes: 47A46, 47B47, 42B30, 30J05, 47B35, 47A15, 30H05 Cite as: arXiv:2606.25649 [math.FA] (or arXiv:2606.25649v2 [math.FA] for this version) https://doi.org/10.48550/arXiv.2606.25649 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Srijan Sarkar [ view email ] [v1] Wed, 24 Jun 2026 10:07:33 UTC (27 KB) [v2] Tue, 25 Aug 2026 07:00:13 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-06 Change to browse by: math math.CV References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Finite-rank commutators of projections onto shift-invariant subspaces#670654
• A note on derivations of Murray-von Neumann algebras#490011
• Differential Inclusions for Gradient and Symmetrized Gradient Operators#1122939
• A characterization of the one-dimensional unimodular projective groups over finite fields#490115
• On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials#1157075