On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><...

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Autores principales:	Marat Pliev, Nonna Dzhusoeva, Ruslan Kulaev
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	bilinear operator narrow operator order-to-norm continuous operator <i>C</i>-compact operator regular operator Köthe–Banach space Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/7efe9e99b3bb4ca9872c2d99128f22db
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spelling	oai:doaj.org-article:7efe9e99b3bb4ca9872c2d99128f22db2021-11-25T18:16:56ZOn Bilinear Narrow Operators10.3390/math92228922227-7390https://doaj.org/article/7efe9e99b3bb4ca9872c2d99128f22db2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2892https://doaj.org/toc/2227-7390In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo lspace="0pt">:</mo><mi>E</mi><mo>×</mo><mi>F</mi><mo>→</mo><mi>W</mi></mrow></semantics></math></inline-formula> defined on the Cartesian product of vector lattices <i>E</i> and <i>F</i> and taking values in a vector lattice <i>W</i> is narrow if the partial operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>y</mi></msub></semantics></math></inline-formula> are narrow for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>E</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>F</mi></mrow></semantics></math></inline-formula>. We prove that, for order-continuous Köthe–Banach spaces <i>E</i> and <i>F</i> and a Banach space <i>X</i>, the classes of narrow and weakly function narrow bilinear operators from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> to <i>X</i> are coincident. Then, we prove that every order-to-norm continuous <i>C</i>-compact bilinear regular operator <i>T</i> is narrow. Finally, we show that a regular bilinear operator <i>T</i> from the Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> of vector lattices <i>E</i> and <i>F</i> with the principal projection property to an order continuous Banach lattice <i>G</i> is narrow if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>\|</mo><mi>T</mi><mo>\|</mo></mrow></semantics></math></inline-formula> is.Marat PlievNonna DzhusoevaRuslan KulaevMDPI AGarticlebilinear operatornarrow operatororder-to-norm continuous operator<i>C</i>-compact operatorregular operatorKöthe–Banach spaceMathematicsQA1-939ENMathematics, Vol 9, Iss 2892, p 2892 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	bilinear operator narrow operator order-to-norm continuous operator <i>C</i>-compact operator regular operator Köthe–Banach space Mathematics QA1-939
spellingShingle	bilinear operator narrow operator order-to-norm continuous operator <i>C</i>-compact operator regular operator Köthe–Banach space Mathematics QA1-939 Marat Pliev Nonna Dzhusoeva Ruslan Kulaev On Bilinear Narrow Operators
description	In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo lspace="0pt">:</mo><mi>E</mi><mo>×</mo><mi>F</mi><mo>→</mo><mi>W</mi></mrow></semantics></math></inline-formula> defined on the Cartesian product of vector lattices <i>E</i> and <i>F</i> and taking values in a vector lattice <i>W</i> is narrow if the partial operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>y</mi></msub></semantics></math></inline-formula> are narrow for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>E</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>F</mi></mrow></semantics></math></inline-formula>. We prove that, for order-continuous Köthe–Banach spaces <i>E</i> and <i>F</i> and a Banach space <i>X</i>, the classes of narrow and weakly function narrow bilinear operators from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> to <i>X</i> are coincident. Then, we prove that every order-to-norm continuous <i>C</i>-compact bilinear regular operator <i>T</i> is narrow. Finally, we show that a regular bilinear operator <i>T</i> from the Cartesian product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> of vector lattices <i>E</i> and <i>F</i> with the principal projection property to an order continuous Banach lattice <i>G</i> is narrow if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>\|</mo><mi>T</mi><mo>\|</mo></mrow></semantics></math></inline-formula> is.
format	article
author	Marat Pliev Nonna Dzhusoeva Ruslan Kulaev
author_facet	Marat Pliev Nonna Dzhusoeva Ruslan Kulaev
author_sort	Marat Pliev
title	On Bilinear Narrow Operators
title_short	On Bilinear Narrow Operators
title_full	On Bilinear Narrow Operators
title_fullStr	On Bilinear Narrow Operators
title_full_unstemmed	On Bilinear Narrow Operators
title_sort	on bilinear narrow operators
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/7efe9e99b3bb4ca9872c2d99128f22db
work_keys_str_mv	AT maratpliev onbilinearnarrowoperators AT nonnadzhusoeva onbilinearnarrowoperators AT ruslankulaev onbilinearnarrowoperators
_version_	1718411390980456448

On Bilinear Narrow Operators

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