p-Summing Bloch mappings on the complex unit disc

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 $1\le p\le \mathrm{\infty }$. It is shown that the linear space of such mappings, equipped with a natural seminorm ${\pi }_p^{\mathbb{B}}$, 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 $\mathbb{D}$ and identify the spaces of normalized $p$-summing Bloch mappings from $\mathbb{D}$ into $X^{\ast }$ under the norm ${\pi }_p^{\mathbb{B}}$ with the duals of such spaces of molecules under the Bloch version of the $p$-Chevet–Saphar tensor norms $d_p$.