On the Ball Covering Property of $\mathbb{L}(\mathbb{X}, \mathbb{Y})$

[2607.10353] On the Ball Covering Property of $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2607.10353 (math) [Submitted on 11 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: On the Ball Covering Property of $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ Authors: Ankan Mishra , Kallol Paul , Debmalya Sain , Shamim Sohel View PDF HTML (experimental) Abstract: We investigate the Ball Covering Property (BCP) of Banach spaces through the lens of Birkhoff-James orthogonality, yielding a new geometric characterization of the property. Applying this framework, we characterize the BCP of the bounded linear operator space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ under specific conditions on $\mathbb{X}$ and $\mathbb{Y}$, proving that an earlier necessary condition is sufficient. As an application, we provide a complete affirmative answer to an open question concerning the BCP of $\mathbb{L}(L^p[0,1])$. We further apply our results to vector-valued Lipschitz spaces $\operatorname{Lip}_0(M,\mathbb{Y})$, obtaining a characterization of the BCP in this setting under the assumption of the Radon-Nikodým property. We also study the stability of the BCP under $p$-norm direct sums. In finite dimensions, we provide a sufficient condition for an $n$-dimensional Banach space to have a minimal covering by $n+1$ balls. Furthermore, we find an upper bound for the minimal ball covering number of $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ in the finite-dimensional setting and prove that this number is exactly $mn+1$ when $\mathbb{X}$ is an $m$-dimensional strictly convex space and $\mathbb{Y}$ is an $n$-dimensional smooth space. Subjects: Functional Analysis (math.FA) MSC classes: Primary 47L05, Secondary 46B20 Cite as: arXiv:2607.10353 [math.FA] (or arXiv:2607.10353v2 [math.FA] for this version) https://doi.org/10.48550/arXiv.2607.10353 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Shamim Sohel [ view email ] [v1] Sat, 11 Jul 2026 15:26:35 UTC (24 KB) [v2] Thu, 27 Aug 2026 08:41:37 UTC (26 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-07 Change to browse by: math 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)

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