p-Summing Bloch mappings on the complex unit disc

Abstract

The notion of p-summing Bloch mapping from the complex unit open disc \(\mathbb {D}\) into a complex Banach space X is introduced for any \(1\le p\le \infty .\) It is shown that the linear space of such mappings, equipped with a natural seminorm \(\pi ^{\mathcal {B}}_p,\) 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 \(\mathbb {D}\) and identify the spaces of normalized p-summing Bloch mappings from \(\mathbb {D}\) into \(X^*\) under the norm \(\pi ^{\mathcal {B}}_p\) with the duals of such spaces of molecules under the Bloch version of the \(p^*\)-Chevet–Saphar tensor norms \(d_{p^*}.\)