An extension of order bounded operators

Document Type : Research Paper

Authors

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

10.22128/gadm.2024.783.1105

Abstract

Let $E$ be a normed lattice and an order dense majorizing sublattice of a vector lattice $E^t$. We extend the norm of $E$ to $E^t$, denoted by $\Vert.\Vert_t$. The pair $(E^t,\Vert.\Vert_t)$ forms a normed lattice and preserves certain lattices and topological properties whenever these properties hold in $E$.
As a consequence, every positive linear operator defined on a normed lattice $E$ has a linear extension to $E^t$. This manuscript provides an explicit formula for these extensions.
The extended operator $T^t$ is a lattice homomorphism from $E^t$ into $F$, and it belongs to $\mathcal{L}_n(E^t,F)$ whenever $0\leq T\in \mathcal{L}_n(E,F)$ and $T(x\wedge y)=Tx \wedge Ty$ for all $0\leq x,y\in E$. Furthermore, if $T\in \mathcal{L}_b(E,F)$ and certain lattice and topological properties hold for $T$, then $T^t\in \mathcal{L}_b(E^t,F)$ will also preserve these properties.

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References

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4. O. van Gaans Seminorms on ordered vector space that extend to Riesz seminorms on large spaces, Indag Mathem, N8, 14(1), 15–30 (2003).
Global Analysis and Discrete Mathematics

Articles in Press, Accepted Manuscript
Available Online from 20 October 2024

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History

  • Receive Date: 23 January 2024
  • Revise Date: 16 August 2024
  • Accept Date: 19 August 2024

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