A Krengel Type Theorem for Compact Operators Between Locally Solid Vector Lattices

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

		Log in

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/g6863-7709-2981-j A Krengel Type Theorem for Compact Operators Between Locally Solid Vector Lattices Zabeti, O. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 3. Abstract: Suppose \(X\) and \(Y\) are locally solid vector lattices. A linear operator \(T:X\to Y\) is said to be \(nb\)-compact provided that there exists a zero neighborhood \(U\subseteq X\), such that \(\overline{T(U)}\) is compact in \(Y\); \(T\) is \(bb\)-compact if for each bounded set \(B\subseteq X\), \(\overline{T(B)}\) is compact. These notions are far from being equivalent, in general. In this paper, we introduce the notion of a locally solid \(AM\)-space as an extension for \(AM\)-spaces in Banach lattices. With the aid of this concept, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices. This extends [1, Theorem 5.7] (established for compact operators between Banach lattices) to different classes of compact operators between locally solid vector lattices. Keywords: compact operator, the Krengel theorem, locally solid \(AM\)-space Language: English Download the full text For citation: Zabeti, O. A Krengel Type Theorem for Compact Operators Between Locally Solid Vector Lattices, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp.76-80. DOI 10.46698/g6863-7709-2981-j + References 1. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Springer, 2006. 2. Zabeti, O. \(AM\)-Spaces from a Locally Solid Vector Lattice Point of View with Applications, Bulletin of the Iranian Mathematical Society, 2021, vol. 47, pp. 1559-1569. DOI: 10.1007/s41980-020-00458-7. 3. Aliprantis, C. D. and Burkinshaw, O. Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, vol. 105, Providence, American Mathematical Society, 2003. 4. Jameson, G. J. O. Topological \(M\)-Spaces, Mathematische Zeitschrift, 1968, vol. 103, pp. 139-150. DOI: 10.1007/BF01110626. 5. Zabeti, O. The Banach–Saks Property from a Locally Solid Vector Lattice Point of View, Positivity, 2021, vol. 25, pp. 1579-1583. DOI: 10.1007/s11117-021-00830-9. 6. Jameson, G. J. O. Ordered Linear Spaces, Lecture Notes in Mathematics, vol. 141, Springer, 1970. 7. Erkursun-Ozcan, N., Anil Gezer, N. and Zabeti, O. Spaces of \(u\tau\)-Dunford-Pettis and \(u\tau\)-Compact Operators on Locally Solid Vector Lattices, Matematicki Vesnik, 2019, vol. 71, no. 4, pp. 351-358. 8. Troitsky, V. G. Spectral Radii of Bounded Operators on Topological Vector Spaces, PanAmerican Mathematical Journal, 2001, vol. 11, no. 3, pp. 1-35. 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


	\| Home \| Editorial board \| Publication ethics \| Peer review guidelines \| Latest issue \| All issues \| Rules for authors \| Online submission system's guidelines \| Submit manuscript \|
	© 1999-2026 Южный математический институт