More on the extension of linear operators on Riesz spaces

Authors

  • 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
https://doi.org/10.15330/cmp.14.2.327-331

Keywords:

positive operator, linear extension, Riesz space, vector lattice
Published online: 2022-07-27

Abstract

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 σ-complete then for every eE+ every positive finitely additive F-valued measure defined on the Boolean algebra Fe of fragments of e has a unique positive linear extension to the ideal Ee of E generated by e. If, moreover, the measure is τ-continuous then the linear extension is order continuous.

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How to Cite
(1)
Fotiy, O.; Gumenchuk, A.; Popov, M. More on the Extension of Linear Operators on Riesz Spaces. Carpathian Math. Publ. 2022, 14, 327-331.