On Composition Ideals and Dual Ideals of Bounded Holomorphic Mappings

Abstract

AbstractApplying a linearization theorem due to Mujica (Trans Am Math Soc 324:867–887, 1991), we study the ideals of bounded holomorphic mappings $$\mathcal {I}\circ \mathcal {H}^\infty $$ I ∘ H ∞ generated by composition with an operator ideal $$\mathcal {I}$$ I . The bounded-holomorphic dual ideal of $$\mathcal {I}$$ I is introduced and its elements are characterized as those that admit a factorization through $$\mathcal {I}^{\textrm{dual}}$$ I dual . For complex Banach spaces E and F, we also analyze new ideals of bounded holomorphic mappings from an open subset $$U\subseteq E$$ U ⊆ E to F such as p-integral holomorphic mappings and p-nuclear holomorphic mappings with $$1\le p<\infty $$ 1 ≤ p < ∞ . We prove that every p-integral (p-nuclear) holomorphic mapping from U to F has relatively weakly compact (compact) range.