Let X and Y be complex Banach spaces, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> be the open unit ball of X and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}L_0(B_X,Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <msub> <mi>L</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the Banach space of all holomorphic Lipschitz maps <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f:B_X\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, endowed with the Lipschitz norm. Given a Banach operator ideal <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, we use the property of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-compactness by Carl and Stephani to introduce and study the subclass of those functions in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {H}L_0(B_X,Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <msub> <mi>L</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for which its Lipschitz image is a relatively <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-compact subset of Y. We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-compact linear operators through linearization/transposition techniques.

$\mathcal {A}$-Compact Holomorphic Lipschitz Mappings on the Unit Ball of a Banach Space

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2026年2月19日

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A. Jiménez-Vargas,
D. Ruiz-Casternado

摘要

Let X and Y be complex Banach spaces, $B_X$ $B_{X}$ be the open unit ball of X and $\mathcal {H}L_0(B_X,Y)$ $H L_{0} (B_{X}, Y)$ be the Banach space of all holomorphic Lipschitz maps $f:B_X\rightarrow Y$ $f : B_{X} \to Y$ such that $f(0)=0$ $f (0) = 0$ , endowed with the Lipschitz norm. Given a Banach operator ideal $\mathcal {A}$ $A$ , we use the property of $\mathcal {A}$ $A$ -compactness by Carl and Stephani to introduce and study the subclass of those functions in $\mathcal {H}L_0(B_X,Y)$ $H L_{0} (B_{X}, Y)$ for which its Lipschitz image is a relatively $\mathcal {A}$ $A$ -compact subset of Y. We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with $\mathcal {A}$ $A$ -compact linear operators through linearization/transposition techniques.

\(\mathcal {A}\)-Compact Holomorphic Lipschitz Mappings on the Unit Ball of a Banach Space

摘要