p-Summing Bloch mappings on the complex unit disc

Abstract

The notion of p-summing Bloch mapping from the complex unit open disc D into a complex Banach space X is introduced for any 1 ≤ p ≤ ∞. It is shown that the linear space of such mappings, equipped with a natural seminorm πpB , is Möbius invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch’s domination/factorization Theorem and the Maurey’s extrapolation Theorem are presented. We also introduce the spaces of X-valued Bloch molecules on D and identify the spaces of normalized p-summing Bloch mappings from D into X∗ under the norm πpB with the duals of such spaces of molecules under the Bloch version of the p∗-Chevet–Saphar tensor norms dp∗ .