Extremal Structure of Convex Sets of Linear Operators on the Space of Continuous Functions

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/s3306-7592-2603-k Extremal Structure of Convex Sets of Linear Operators on the Space of Continuous Functions Tamaeva, V. A. , Tasoev, B. B. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1. Abstract: The goal of this paper is to describe the extreme points of a convex set of linear positive operators acting from the space of continuous real-valued functions on a compact set to an order-complete vector lattice and mapping the identity unit to some fixed nonzero element. The main tool of our study is the canonical sublinear operator method proposed by S. S. Kutateladze. The idea of this method is that an arbitrary sublinear operator can be represented as the composition of a linear operator and a specific sublinear operator, called the canonical Kutateladze sublinear operator. The extreme points of an arbitrary sublinear operator are the composition of the linear operator and the extreme points of the canonical Kutateladze sublinear operator. Using this fact, we obtained a description of the extreme points of the convex set of positive linear operators under study using lattice homomorphisms, in particular, pure states, which represent a special type of extreme points of the canonical Kutateladze sublinear operator. Keywords: vector lattice, extreme point, lattice homomorphism, quasi-regular measure, sublinear operator Fund name: The work was carried out at the North Caucasus Center for Mathematical Research of the Russian Academy of Sciences with the support of the Ministry of Education and Science of Russia, agreement No. 075-02-2026-738. Language: Russian Download the full text Download JATS XML For citation: Tamaeva, V. A. and Tasoev, B. B. Extremal Structure of Convex Sets of Linear Operators on the Space of Continuous Functions, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 134-144 (in Russian). DOI 10.46698/s3306-7592-2603-k + References 1. Kutateladze, S. S. Extreme Points of Subdifferentials, Doklady Akademii Nauk SSSR, 1978, vol. 242, no. 5, pp. 1001-1003 (in Russian). 2. Kutateladze, S. S. The Krein-Mil'man Theorem and Its Inverse, Siberian Mathematical Journal, 1980, vol. 21, no. 1, pp. 97-103. DOI: 10.1007/BF00970127. 3. Kutateladze, S. S. Caps and Faces of Sets of Operators, Doklady Akademii Nauk SSSR, 1985, vol. 280, no. 2, pp. 285-283 (in Russian). 4. Kutateladze, S. S. Criteria for Subdifferentials that Represent Caps and Faces, Siberian Mathematical Journal, 1986, vol.27, pp. 417-423. DOI: 10.1007/BF00969278. 5. Kutateladze, S. S. Support Sets for Sublinear Operators, Doklady Akademii Nauk SSSR, 1977, vol. 17, no. 5, pp. 1428-1431 (in Russian). 6. Tamaeva, V. A. and Tasoev, B. B. A Note on the Representation of Lattice Homomorphisms, Positivity, 2024, vol. 28, no. 5. DOI: 10.1007/s11117-024-01095-8. 7. Kusraev, A. G. Dominated Operators, Kluwer, Springer, 2000. 8. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, London, Acad. Press Inc., 1985, xx+376 p. 9. Wright, M. Stone-Algebra-Valued Measures and Integrals, Proceedings of the London Mathematical Society, 1969, vol. 19, no. 3, pp. 107-122. 10. Kusraev, A. G. and Tasoev, B. B. Kantorovich-Wright Integration and Representation of Vector Lattices, Journal of Mathematical Analysis and Applications, 2017, vol. 455, no. 1, pp. 554-568. DOI: 10.1016/j.jmaa.2017.05.059. 11. Kusraev, A. G. and Kutateladze, S. S. Subdifferentials: Theory and Applications, Springer Dordrecht, 2012, ix+405 p. DOI: 10.1007/978-94-011-0265-0. 12. Kusraev, A. G. and Tasoev, B. B. Boolean Valued Analysis: Selected Topics, Vladikavkaz, Southern Mathematical Institute VSC RAS and RNO-A, 2014, iv+400 p. DOI: 10.13140/RG.2.1.2164.6486. ← Contents of issue


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