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

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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	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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. ← Contents of issue


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