p-Summing Bloch mappings on the complex unit disc

Abstract

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