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 $1\leq p\leq\infty$. It is shown that the linear space of such mappings, equipped with a natural seminorm $\pi^{ \mathbb{ B}}_p$, is M\"obius-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 $\pi^{ \mathbb{ B}}_p$ with the duals of such spaces of molecules under the Bloch version of the $p$-Chevet--Saphar tensor norms $d_p$.
Categories: math.FA math.CV