On the (p, q)-compactness of Banach-valued Bloch mappings

This work develops a comprehensive theory of what we call (p, q)-compact Bloch mappings, extending to the Bloch setting the concept of (p, q)-compact linear operators introduced by Ain, Lillemets, and Oja. We introduce and deeply study holomorphic mappings from the open unit disc in the complex plane in a complex Banach space, whose measures of the size of (p, q)-compactness associated with the Bloch range are finite. The collection of all zero-preserving (p, q)-compact Bloch mappings, endowed with a suitable norm, is shown to constitute a surjective s-Banach ideal of normalized Bloch mappings, where the exponent s is given by s = pq/(p + q). This ideal becomes regular for the collection of all reflexive complex Banach spaces.

This paper establishes several structural properties of these mappings. We prove invariance under Möbius transformations and a linearization theorem that identifies a Bloch mapping as (p, q)-compact precisely when its associated continuous linear operator on the Bloch-free space is (p, q)-compact. This correspondence allows us to extend many classical results on operator ideals to the nonlinear Bloch framework. We also obtain factorization theorems for (p, q)-compact Bloch mappings through compositions involving compact Bloch mappings and (p, q)-compact linear operators. Furthermore, we introduce and analyze the subclass of (t, u, v)-nuclear Bloch mappings, providing characterizations and their ideal structure parallel to those known in operator theory.

Hence, this study unifies and generalizes previous results on compact and p-compact Bloch mappings, establishing a deep interaction between operator ideals and the geometry of holomorphic mappings on the open complex unit disc, and demonstrating the robustness of the (p, q)-compact setting under linearization, factorization, and Möbius invariance.