On Equi-Continuity of Bounded on Order Intervals Families of Semi-Norms

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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/q3483-1455-3369-u On Equi-Continuity of Bounded on Order Intervals Families of Semi-Norms Gorokhova, S. G. , Emelyanov, E. Yu. , Storozhuk, K. V. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 2. Abstract: We investigate conditions which provide equi-continuity of bounded on order intervals families of semi-norms. This investigation develops further recent results on uniform boundedness of bounded on order intervals families of linear operators from an ordered Banach space with a closed generating cone to a normed space. More precisely, we prove that a family of semi-norms on an ordered Banach space with a closed generating cone is equi-continuous if it is equi-bounded on order intervals. In particular, every semi-norm on an ordered Banach space with a closed generating cone is continuous, whenever it is bounded on order intervals. It leads to straightforward proofs of above-mentioned recent results on automatic continuity of certain linear operators and on uniform boundedness of some families of linear operators. Furthermore, we prove that every order-to-topology bounded set of linear operators from an ordered Banach space with a closed generating cone to a locally convex space is equi-continuous. Keywords: ordered Banach space, order interval, equi-continuous family of semi-norms. Funding: The work of the second author was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF-2026-0022). Language: English Download the full text Download JATS XML For citation: Gorokhova, S. G., Emelyanov, E. Y. and Storozhuk, K. V. On Equi-Continuity of Bounded on Order Intervals Families of Semi-Norms, Vladikavkaz Math. J., 2025, vol. 28, no. 2, pp. 35-39. DOI 10.46698/q3483-1455-3369-u + References 1. Abramovich, Y. A. and Aliprantis, C. D. An Invitation to Operator Theory, Graduate Studies in Mathematics, vol. 50, Providence, RI, American Mathematical Society, 2002, 530 p. 2. Montgomery-Smith, S. and Semenov, E. M. Averaging of Fourier-Haar Coefficients, Sbornik: Mathematics, 1999, vol. 190, no. 10, pp. 1449-1463. DOI: 10.1070/SM1999v190n10ABEH000432. 3. Gorokhova, S. On Compact (Limited) Operators Between Hilbert and Banach Spaces, Filomat, 2024, vol. 38, no. 33, pp. 11633-11637. DOI: 10.2298/FIL2433633G. 4. Emelyanov, E., Erkursun-Ozcan, N. and Gorokhova, S. $d$-Operators in Banach Lattices, Siberian Mathematical Journal, 2025, vol. 66, no. 6, pp. 1499-1508. DOI: 10.1134/S003744662506014X. 5. Emelyanov, E. Collective Order Convergence and Collectively Qualified Sets of Operators, Siberian Electronic Mathematical Reports, 2026, vol. 23, no. 1, pp. 289-297. DOI: 10.33048/semi.2026.23.018. 6. Emelyanov, E., Erkursun-Ozcan, N. and Gorokhova, S. Collective Order Boundedness of Sets of Operators Between Ordered Vector Spaces, Results in Mathematics, 2025, vol. 80, article no. 70. DOI: 10.1007/s00025-025-02386-6. 7. Krein, S. G., Petunin, Yu. I. and Semenov, E. M. Interpolation of Linear Operators, Translation of Mathematical Monographs, vol. 54, Providence, RI, American Mathematical Society, 2002. 8. Aliprantis, C. D. and Tourky, R. Cones and Duality, Graduate Studies in Mathematics, vol. 84, Providence, RI, American Mathematical Society, 2007. 9. Casini, E., Miglierina, E., Polyrakis, I. A. and Xanthos, F. Reflexive Cones, Positivity, 2013, vol. 17, pp. 911-933. DOI: 10.1007/s11117-012-0212-6. 10. Mironov, A. V. Some Properties of Partially Ordered Topological Vector Spaces and Algebras, Russian Mathematical Surveys, 1979, vol. 34, no. 6, pp. 184-187. DOI: 10.1070/RM1979v034n06ABEH003332. 11. Emelyanov, E. Automatic Boundedness of Some Operators Between Ordered and Topological Vector Spaces, Vladikavkaz Mathematical Journal, 2026, vol. 28, no. 1, pp. 62-67. DOI: 10.46698/i3984-2243-2985-z. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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