Abstract
The span of positive linear operators belonging to an operator linear class P and acting between Banach lattices is rarely a Banach space under the operator norm. We investigate the enveloping norm \(\Vert S\Vert _{\text {r-P}}=\inf \{\Vert T\Vert : \pm S\le T\in \text {P}\}\) on \({\text {span}}(\text {P}_+(E,F))\) that is complete under rather mild assumptions on P.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)
Alpay, S., Emelyanov, E., Gorokhova, S.: \(o\tau \)-Continuous, Lebesgue, KB, and levi operators between vector lattices and topological vector spaces topological vector spaces. Results Math. 77(117), 25 (2022)
Aqzzouz, B., Elbour, A.: Some characterizations of almost Dunford-Pettis operators and applications. Positivity 15, 369–380 (2011)
Bouras, K., Lhaimer, D., Moussa, M.: On the class of almost L-weakly and almost M-weakly compact operators. Positivity 22, 1433–1443 (2018)
Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 119, 55–58 (1984)
Chen, Z.L., Wickstead, A.W.: Incompleteness of the linear span of the positive compact operators. Proc. Am. Math. Soc. 125, 3381–3389 (1997)
Chen, Z.L., Wickstead, A.W.: The order properties of r-compact operators on Banach lattices. Acta Math. Sin. Engl. Ser. 23(3), 457–466 (2007)
Cheng, N.: Dedekind \(\sigma \)-complete vector lattice of b-AM-compact operators. Quaest. Math. 40(3), 313–318 (2017)
Emelyanov, E.: On KB and Levi operators in Banach lattices. arxiv.org/abs/2312.05685v2
Gorokhova, S.G., Emelyanov, E.: On operators dominated by the Kantorovich-Banach and Levi operators in locally solid lattices. Vladikavkaz Math. J. 24(3), 55–61 (2022)
Galindo, P., Miranda, V.C.C.: Grothendieck-type subsets of Banach lattices. J. Math. Anal. Appl. 506(1), 125570 (2022)
Lhaimer, D., Bouras, K., Moussa, M.: On the class of order L-weakly and order M-weakly compact operators. Positivity 25, 1569–1578 (2021)
Krengel, U.: Remark on the modulus of compact operators. Bull. Am. Math. Soc. 72, 132–133 (1966)
Meyer-Nieberg, P.: Banach Lattices. Universitext Springer-Verlag, Berlin (1991)
Sanchez, J. A.: Operators on Banach lattices. Ph. D. Thesis, Complutense University, Madrid, (1985)
Wang, Z., Chen, Z.: Un L- and M-weakly compact operators on Banach lattices. Positivity 26, 43 (2022)
Funding
The research of the second author was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF-2022–0004).
Author information
Authors and Affiliations
Contributions
All authors have contributed equally to this work for writing, review and editing. All authors have read and agreed to the published version of the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
None of the authors has Conflict of interest of a financial or personal nature.
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Alpay, S., Emelyanov, E. & Gorokhova, S. Enveloping norms of regularly P-operators in Banach lattices. Positivity 28, 37 (2024). https://doi.org/10.1007/s11117-024-01055-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11117-024-01055-2