Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1< p_1, \ldots , p_n< \infty , 1\le q < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo><</mo> <mi>∞</mi> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> be such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sum \limits _{i=1}^n \frac{1}{p_i} < \frac{1}{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> <mo><</mo> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu _1, \ldots , \mu _n, \nu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>ν</mi> </mrow> </math></EquationSource> </InlineEquation> be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive n-linear operators from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell _{p_1} \times \cdots \times \ell _{p_n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>ℓ</mi> <msub> <mi>p</mi> <mi>n</mi> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_q(\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and from <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_{p_1}(\mu _1) \times \cdots \times L_{p_n}(\mu _n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>L</mi> <msub> <mi>p</mi> <mi>n</mi> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell _{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> are compact. This result, along with other related ones concerning free Banach lattices, shall emerge as consequences of some facts we prove about M-weakly compact multilinear operators on Banach lattices.

Compact positive multilinear operators on Banach lattices

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2026年3月2日

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Geraldo Botelho,
Vinícius C. C. Miranda

摘要

Let $1< p_1, \ldots , p_n< \infty , 1\le q < \infty $ $1 < p_{1}, \dots, p_{n} < \infty, 1 \leq q < \infty$ be such that $\sum \limits _{i=1}^n \frac{1}{p_i} < \frac{1}{q}$ $\sum_{i = 1}^{n} \frac{1}{p_{i}} < \frac{1}{q}$ and let $\mu _1, \ldots , \mu _n, \nu $ $μ_{1}, \dots, μ_{n}, ν$ be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive n-linear operators from $\ell _{p_1} \times \cdots \times \ell _{p_n}$ $ℓ_{p_{1}} \times \dots \times ℓ_{p_{n}}$ to $L_q(\nu )$ $L_{q} (ν)$ and from $L_{p_1}(\mu _1) \times \cdots \times L_{p_n}(\mu _n)$ $L_{p_{1}} (μ_{1}) \times \dots \times L_{p_{n}} (μ_{n})$ to $\ell _{q}$ $ℓ_{q}$ are compact. This result, along with other related ones concerning free Banach lattices, shall emerge as consequences of some facts we prove about M-weakly compact multilinear operators on Banach lattices.