Completely continuous multilinear mappings on 𝐿₁

Proceedings of the American Mathematical Society, Series B Pub Date : 2024-05-15 DOI:10.1090/bproc/213

Raffaella Cilia, Joaquín Gutiérrez

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Bourgain states that, given a Banach space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, an operator <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T colon upper L 1 left-bracket 0 comma 1 right-bracket right-arrow upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>T</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">T:L_1[0,1]\\to X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is completely continuous if and only if its composition with the natural inclusion <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i Subscript normal infinity Baseline colon upper L Subscript normal infinity Baseline left-bracket 0 comma 1 right-bracket right-arrow upper L 1 left-bracket 0 comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>i</mml:mi>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:msub>\n <mml:mo>:</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">i_\\infty :L_\\infty [0,1] \\to L_1[0,1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is compact. We extend this result to multilinear mappings on products of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L 1 left-bracket 0 comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L_1[0,1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> spaces, and consider also the composition with the natural inclusion <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i colon upper C left-bracket 0 comma 1 right-bracket right-arrow upper L 1 left-bracket 0 comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>i</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mi>C</mml:mi>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">i:C[0,1]\\to L_1[0,1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We show that a multilinear mapping on a product of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L 1 left-bracket 0 comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L_1[0,1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"136 38","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/213","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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Abstract

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X X , an operator T : L 1 [ 0 , 1 ] → X T:L_1[0,1]\to X is completely continuous if and only if its composition with the natural inclusion i ∞ : L ∞ [ 0 , 1 ] → L 1 [ 0 , 1 ] i_\infty :L_\infty [0,1] \to L_1[0,1] is compact. We extend this result to multilinear mappings on products of L 1 [ 0 , 1 ] L_1[0,1] spaces, and consider also the composition with the natural inclusion i : C [ 0 , 1 ] → L 1 [ 0 , 1 ] i:C[0,1]\to L_1[0,1] . We show that a multilinear mapping on a product of L 1 [ 0 , 1 ] L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range.

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𝐿₁上的完全连续多线性插值

罗森塔尔（H. Rosenthal）和布尔甘（J. Bourgain）的一个有用结果指出，给定一个巴拿赫空间 X X，当且仅当一个算子 T : L 1 [ 0 , 1 ] → X T:L_1[0,1]\to X 与自然包含 i ∞ : L ∞ [ 0 , 1 ] → L 1 [ 0 , 1 ] i_\infty :L_\infty [0,1] \to L_1[0,1] 的组合是紧凑的时候，这个算子 T 才是完全连续的。我们将这一结果扩展到 L 1 [ 0 , 1 ] L_1[0 , 1] 空间乘积上的多线性变换，并考虑与自然包含 i : C [ 0 , 1 ] → L 1 [ 0 , 1 ] i:C[0 , 1]\to L_1[0 , 1] 的组合。我们证明，L 1 [ 0 , 1 ] L_1[0,1]空间乘积上的多线性映射是完全连续的，当且仅当其相关的多度量具有相对规范紧凑的范围。

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Proceedings of the American Mathematical Society, Series B

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