On quotients of ideals of weighted holomorphic mappings
Belacel Amar, Bougoutaia Amar, Rueda Pilar
TL;DR
The paper develops a weighted holomorphic analogue of the left-hand quotient of operator ideals and analyzes when these quotients reproduce the whole space $\mathcal{H}_{v}^{\infty }$. It defines $\mathcal{A}^{-1}\circ \mathcal{I}^{\mathcal{H}_{v}^{\infty }}$ with a natural norm and shows that, if the weighted holomorphic ideal $\mathcal{I}^{\mathcal{H}_{v}^{\infty }}$ has the linearization property (LP) in $\mathcal{A}$, then $\mathcal{A}^{-1}\circ \mathcal{I}^{\mathcal{H}_{v}^{\infty }}=\mathcal{H}_{v}^{\infty }$, with corresponding norm equality. The results recover classical coincidences for several operator ideals (e.g., finite rank, approximable, compact, weakly compact and their $p$-variants, right $p$-nuclear) and relate left-hand quotients to composition-ideals. In nontrivial examples, the paper analyzes Grothendieck and Rosenthal weighted holomorphic mappings, proving that $\mathcal{H}_{v\mathcal{O}}^{\infty }\cong \mathcal{O}(\mathcal{G}_{v}^{\infty }(U),F)$ with $\mathcal{H}_{v\mathcal{O}}^{\infty }=\mathcal{O}^{-1}\circ \mathcal{H}_{v}^{\infty }$, and $\mathcal{H}_{v\mathcal{R}}^{\infty }=\mathcal{V}^{-1}\circ \mathcal{H}_{v\mathcal{K}}^{\infty }$, thereby embedding these classical operator-ideal notions into the weighted holomorphic setting.
Abstract
We explore the procedure given by left-hand quotients in the context of weighted holomorphic ideals. On the one hand, we show that this procedure does not generate new ideals other than the ideal of weighted holomorphic mappings when considering the left-hand quotients induced by the ideals of $p$-compact, weakly $p$-compact, unconditionally $p$-compact, approximable or right $p$-nuclear operators with their respective weighted holomorphic ideals. On the other hand, the procedure is of interest when considering other operators ideals as it provides new weighted holomorphic ideals. This is the case of the ideal of Grothendieck weighted holomorphic mappings or the ideal of Rosenthal weighted holomorphic mappings, where the applicability of this construction is shown.