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<p>The notion of a Levi operator is an operator abstraction of the Levi property of a norm or, more generally,of the Levi topology on a locally solid vector lattice.Various aspects of Levi operators have been studied recently by several authors. The present paperis devoted to Levi operators from a normed lattice to a vector lattice. It is proved that every continuous finite-rank operatoris a&#xa0;Levi operator. An example is given showing that the sum of a positive rank one operatorand a&#xa0;positive compact Levi operator need not be a Levi operator.We prove that every quasi Levi operator is&#xa0;continuous.It is shown that the set of Levi operators on the space of convergent sequences is not complete in the operator norm.Several results concerning the domination problem for Levi operators and the relations between Levi operators and KB-spacesare established.</p>

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On Levi Operators Between Normed and Vector Lattices

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • E. Yu. Emelyanov

摘要

The notion of a Levi operator is an operator abstraction of the Levi property of a norm or, more generally,of the Levi topology on a locally solid vector lattice.Various aspects of Levi operators have been studied recently by several authors. The present paperis devoted to Levi operators from a normed lattice to a vector lattice. It is proved that every continuous finite-rank operatoris a Levi operator. An example is given showing that the sum of a positive rank one operatorand a positive compact Levi operator need not be a Levi operator.We prove that every quasi Levi operator is continuous.It is shown that the set of Levi operators on the space of convergent sequences is not complete in the operator norm.Several results concerning the domination problem for Levi operators and the relations between Levi operators and KB-spacesare established.