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

Year 2024, Volume: 7 Issue: 3, 98 - 113, 15.09.2024

Antonio Jiménez Vargas , David Ruiz Casternado

https://doi.org/10.33205/cma.1518651

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

Ethical Statement

This is an original paper and we cite all the necessary references.

Supporting Institution

This research has been supported in part by grant PID2021-122126NB-C31 funded by MCIN/AEI/ 10.13039/501100011033 and by ``ERDF A way of making Europe'', and by Junta de Andalucía grant FQM194.

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