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

Authors : Antonio Jiménez Vargas, David Ruiz Casternado

Pages : 98-113

Doi:10.33205/cma.1518651

View : 70 | Download : 79

Publication Date : 2024-09-15

Article Type : Research Paper

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