New ideals of Bloch mappings which are I-factorizable and Möbius-invariant

Abstract

In this paper, we introduce an unified method for generating ideals of Möbius-invariant Banach-valued Bloch mappings on the complex open unit disc $\D$, through the composition with the members of a Banach operator ideal $\I$. Using linearisation of derivatives of Banach-valued normalized Bloch mappings on $\D$, this composition method yields the so-called ideals of $\I$-factorizable normalized Bloch mappings $\I\circ\hat{\B}$, where $\hat{\B}$ denotes the class of normalized Bloch mappings on $\D$. We present new examples of them as ideals of separable (Rosenthal, Asplund) normalized Bloch mappings and $p$-integral (strictly $p$-integral, $p$-nuclear) normalized Bloch mappings for any $p\in[1,\infty)$. Moreover, the Bloch dual ideal $\I^{\hat{\B}\text{-}\d}$ of an operator ideal $\I$ is introduced and shown that it coincides with the composition ideal $\I^\d\circ\hat{\B}$.

Keywords

• Bloch mapping
• linearisation
• factorization theorems
• operator ideal

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