On unbounded order continuous operators


TURAN B., ALTIN B., Gürkök H.

Turkish Journal of Mathematics, vol.46, no.8, pp.3391-3399, 2022 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 46 Issue: 8
  • Publication Date: 2022
  • Doi Number: 10.55730/1300-0098.3339
  • Journal Name: Turkish Journal of Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.3391-3399
  • Keywords: Order convergence, Riesz space, Unbounded order continuous dual, Unbounded order continuous operator, Unbounded order convergence
  • Gazi University Affiliated: Yes

Abstract

© TÜBİTAKLet U and V be two Archimedean Riesz spaces. An operator S: (Formula Presented) is said to be unbounded order continuous (uo-continuous), if (Formula Presented) in U implies (Formula Presented). In this paper, we give some properties of the uo-continuous dual (Formula Presented) of U. We show that a nonzero linear functional f on U is uo-continuous if and only if f is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the uo-continuous dual (Formula Presented). In general, by giving an example that the uo-continuous dual (Formula Presented) is not a band in (Formula Presented), we obtain the conditions for the uo-continuous dual of a Banach lattice U to be a band in (Formula Presented). Then, we examine the properties of uo-continuous operators. We show that S is an order continuous operator if and only if S is an unbounded order continuous operator when S is a lattice homomorphism between two Riesz spaces U and V. Finally, we proved that if an order bounded operator S: U → V between Archimedean Riesz space U and atomic Dedekind complete Riesz space V is uo-continuous, then |S| is uo-continuous.