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: | , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/7efe9e99b3bb4ca9872c2d99128f22db |
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| Sumario: | 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. |
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