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

Abstract Let X and Y be complex Banach spaces, $$B_X$$ 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 → 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.