More on the extension of linear operators on Riesz spaces

  • O.G. Fotiy Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, Ukraine
  • A.I. Gumenchuk Bukovinian State Medical University, 2 Teatralna sq., 58002, Chernivtsi, Ukraine
  • M.M. Popov Pomeranian University in Słupsk, 76-200, Słupsk, Poland, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: positive operator, linear extension, Riesz space, vector lattice
Published online: 2022-07-27


The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone $E^+$ of a Riesz space $E$ taking values in an Archimedean Riesz space $F$, to the entire space $E$. We prove that, if $E$ has the principal projection property and $F$ is Dedekind $\sigma$-complete then for every $e \in E^+$ every positive finitely additive $F$-valued measure defined on the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ has a unique positive linear extension to the ideal $E_e$ of $E$ generated by $e$. If, moreover, the measure is $\tau$-continuous then the linear extension is order continuous.

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How to Cite
Fotiy O., Gumenchuk A., Popov M. More on the Extension of Linear Operators on Riesz Spaces. Carpathian Math. Publ. 2022, 14 (2), 327-331.