Disjointness Preserving Operators on Complex Riesz Spaces View Full Text


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Article Info

DATE

1997-06

AUTHORS

J. J. Grobler, C. B. Huijsmans

ABSTRACT

It is proven that ifE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$_\mathbb{C} $$ \end{document} and F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$_\mathbb{C} $$ \end{document} are complex Riesz spaces and ifT is an order bounded disjointness preserving operator fromE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$_\mathbb{C} $$ \end{document} intoF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$_\mathbb{C} $$ \end{document} , then\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$|Tz| = |T|z|| for all z \in E_\mathbb{C} . $$ \end{document}This fundamental result of M. Meyer is obtained by elementary means using as the main tool the functional calculus derived from the Freudenthal spectral theorem. It is also shown that ifT is an order bounded disjointness preserving operator, a formula of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Tz = sgn T\left( {|z|} \right) for all z \in E_\mathbb{C} $$ \end{document}holds. It implies a polar decomposition of an order bounded disjointness preserving operator as the product of a Riesz homomorphism and an orthomorphism. Results of P. Meyer-Nieberg in this regard are generalized. More... »

PAGES

155-164

References to SciGraph publications

  • 1991. Lattice-Ordered Algebras and f-Algebras: A Survey in POSITIVE OPERATORS, RIESZ SPACES, AND ECONOMICS
  • 1995. Diagonals of the Powers of an Operator on a Banach Lattice in OPERATOR THEORY IN FUNCTION SPACES AND BANACH LATTICES
  • 1993-05. Abstract multiplication semigroups in MATHEMATISCHE ZEITSCHRIFT
  • Journal

    TITLE

    Positivity

    ISSUE

    2

    VOLUME

    1

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1023/a:1009746711470

    DOI

    http://dx.doi.org/10.1023/a:1009746711470

    DIMENSIONS

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