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

Abstract Let X and Y be complex Banach spaces, $$B_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> be the open unit ball of X and $$\mathcal {H}L_0(B_X,Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the Banach space of all holomorphic Lipschitz maps $$f:B_X\rightarrow Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> such that $$f(0)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , endowed with the Lipschitz norm. Given a Banach operator ideal $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> , we use the property of $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -compactness by Carl and Stephani to introduce and study the subclass of those functions in $$\mathcal {H}L_0(B_X,Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for which its Lipschitz image is a relatively $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -compact subset of Y . We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -compact linear operators through linearization/transposition techniques.