2024/08/26 by M. G. Cabrera-Padilla, Cabrera-Padilla, M. G., A. Jiménez-Vargas +3 · 1 citation
Mathematics · #Analytic and geometric function theory #Functional Equations Stability Results #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2408.14459
Given an open subset U of a complex Banach space E, a weight v on U, and a complex Banach space F, let H^∞v(U,F) denote the Banach space of all weighted holomorphic mappings f\colon U→ F, under the weighted supremum norm ‖f‖v:=sup\v(x)‖f(x)‖\colon x∈ U\. In this paper, we introduce and study the classes of weighted holomorphic mappings H^∞_vKp(U,F) (resp., H^∞_vKwp(U,F) and H^∞_vKup(U,F)) for which the set (vf)(U) is relatively p-compact (resp., relatively weakly p-compact and relatively unconditionally p-compact). We prove that these mapping classes are characterized by p-compact (resp., weakly p-compact and unconditionally p-compact) linear operators defined on a Banach predual space of H^∞v(U) by linearization. We show that H^∞_vKp (resp., H^∞_vKwp and H^∞_vKup) is a Banach ideal of weighted holomorphic mappings which is generated by composition with the ideal of p-compact (resp., weakly p-compact and unconditionally p-compact) linear operators and contains the Banach ideal of all right p-nuclear weighted holomorphic mappings. We also prove that these weighted holomorphic mappings can be factorized through a quotient space of lp^*, and f\inH^∞_vKp(U,F) (resp., f\inH^∞_vKup(U,F)) if and only if its transposition ft is quasi p-nuclear (resp., quasi unconditionally p-nuclear).