Unbounded Order-to-Order Continuous Operators on Riesz Spaces

Document Type : Research Article

Authors

Department of Mathematics and Application Faculty of Sciences University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

‎Let $E$ and $F$ be two Riesz spaces‎. ‎An operator $T \colon E\rightarrow F$ between two Riesz spaces is said to be‎ ‎unbounded order-to-order continuous whenever $x _\alpha \xrightarrow{uo}0$ in $E$ implies $Tx _\alpha \xrightarrow{o}0$ in $F$ for each net $(x_\alpha)\subseteq E$‎. ‎This paper aims to investigate several properties of a novel class of operators and their connections to established operator classifications‎. ‎Furthermore‎, ‎we introduce a new class of operators‎, ‎which we refer to as order-to-unbounded order continuous operators‎. ‎An operator $T \colon E\rightarrow F$ between two Riesz spaces is said to be‎ ‎order-to-unbounded order continuous (for short‎, ‎$ouo$-continuous)‎, ‎if $x _\alpha \xrightarrow{o}0$ in $E$ implies $Tx _\alpha \xrightarrow{uo}0$ in $F$ for each net $(x_\alpha)\subseteq E$‎. ‎In this manuscript‎, ‎we investigate the lattice properties of a certain class of objects and demonstrate that‎, ‎under certain conditions‎, ‎order continuity is equivalent to unbounded order-to-order continuity of operators on Riesz spaces‎. ‎Additionally‎, ‎we establish that the set of all unbounded order-to-order continuous linear functionals on a Riesz space $E$ forms a band of $E^\sim$.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 8, Issue 1
July 2023
Pages 82-90

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History

  • Receive Date: 04 August 2023
  • Revise Date: 06 January 2024
  • Accept Date: 21 January 2024
  • Publish Date: 26 January 2024

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