The injective hull of ideals of weighted holomorphic mappings

Inspired by the interpolative ideal procedure for linear operators due to Matter, the concept of interpolative ideals of a Banach normalized Bloch ideal is introduced. For σ ∈ [0, 1), we prove that the generated ideal is an injective Banach normalized Bloch ideal which is located between the injective hull and the closed injective hull of . We apply this interpolative procedure to normalized Bloch ideals generated by composition and duality. In particular, normalized Bloch ideals generated by composition with p‐summing operator ideals are characterized in terms of a Pietsch‐type domination property and a summability property.

The notion of p-compact sets arises naturally from Grothendieckʼs characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm κp). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ϵ-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the κp-approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010).

The concept of injective hull of ideals of normalized Bloch mappings is introduced and a characterization is established in terms of a domination property. The injective hulls of normalized Bloch ideals generated by the procedures of composition and duality are described and applied to concrete examples of normalized Bloch ideals. A Bloch variant of a known characterization due to Jarchow and Pelczyński for the closed injective hull of an operator ideal is stated in terms of an Ehrling-type inequality.