A-Compact Holomorphic Lipschitz Mappings on the Unit Ball of a Banach Space

Let X and Y be complex Banach spaces, BX B X be the open unit ball of X and HL₀(BX,Y) H L 0 ( B X , Y ) be the Banach space of all holomorphic Lipschitz maps f:BX→ Y f : B X → Y such that $$f(0)=0$$ f ( 0 ) = 0 , endowed with the Lipschitz norm. Given a Banach operator ideal A A , we use the property of A A -compactness by Carl and Stephani to introduce and study the subclass of those functions in HL₀(BX,Y) H L 0 ( B X , Y ) for which its Lipschitz image is a relatively 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 A A -compact linear operators through linearization/transposition techniques.