Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, ..., <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$V_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> be non-null bounded multilinear operators, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ T:Y_{1}\times \cdot \cdot \cdot \times Y_{k}\rightarrow Z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>×</mo> <mo>·</mo> <mo>·</mo> <mo>·</mo> <mo>×</mo> <msub> <mi>Y</mi> <mi>k</mi> </msub> <mo stretchy="false">→</mo> <mi>Z</mi> </mrow> </math></EquationSource> </InlineEquation> a bounded multilinear operator with the property that there exists <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \left\| T\left( y_{1},...,y_{k}\right) \right\| \ge m\left\| y_{1}\right\| \cdot \cdot \cdot \left\| y_{k}\right\| $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="∥" open="∥"> <mi>T</mi> <mfenced close=")" open="("> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> </mfenced> </mfenced> <mo>≥</mo> <mi>m</mi> <mfenced close="∥" open="∥"> <msub> <mi>y</mi> <mn>1</mn> </msub> </mfenced> <mo>·</mo> <mo>·</mo> <mo>·</mo> <mfenced close="∥" open="∥"> <msub> <mi>y</mi> <mi>k</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \left( y_{1},...,y_{k}\right) \in Y_{1}\times \cdot \cdot \cdot \times Y_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> </mfenced> <mo>∈</mo> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>×</mo> <mo>·</mo> <mo>·</mo> <mo>·</mo> <mo>×</mo> <msub> <mi>Y</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$T\circ \left( V_{1},...,V_{k}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∘</mo> <mfenced close=")" open="("> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is strongly p-summing (resp. p-dominated) if and only if all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$V_{i}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> are strongly p-summing (resp. p-dominated). Various concrete examples are given.

A way to construct strongly summing and dominated multilinear operators

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2026年7月22日

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Dumitru Popa

摘要

Let $V_{1}$ $V_{1}$ , ..., $V_{k}$ $V_{k}$ be non-null bounded multilinear operators, $ T:Y_{1}\times \cdot \cdot \cdot \times Y_{k}\rightarrow Z$ $T : Y_{1} \times \cdot \cdot \cdot \times Y_{k} \to Z$ a bounded multilinear operator with the property that there exists $m>0$ $m > 0$ such that $ \left\| T\left( y_{1},...,y_{k}\right) \right\| \ge m\left\| y_{1}\right\| \cdot \cdot \cdot \left\| y_{k}\right\| $ $∥T, (y_{1}, . . ., y_{k})∥ \geq m ∥y_{1}∥ \cdot \cdot \cdot ∥y_{k}∥$ for all $ \left( y_{1},...,y_{k}\right) \in Y_{1}\times \cdot \cdot \cdot \times Y_{k}$ $(y_{1}, . . ., y_{k}) \in Y_{1} \times \cdot \cdot \cdot \times Y_{k}$ and $1\le p<\infty $ $1 \leq p < \infty$ . We prove that $T\circ \left( V_{1},...,V_{k}\right) $ $T \circ (V_{1}, . . ., V_{k})$ is strongly p-summing (resp. p-dominated) if and only if all $V_{i}$ $V_{i}$ are strongly p-summing (resp. p-dominated). Various concrete examples are given.