On quotients of ideals of weighted holomorphic mappings

TL;DR

This paper investigates how left-hand quotient procedures affect weighted holomorphic ideals, showing they do not create new ideals for certain operator classes but do for others like Grothendieck and Rosenthal mappings.

Contribution

It demonstrates that left-hand quotient procedures do not generate new ideals for specific classes but can produce new weighted holomorphic ideals for others, such as Grothendieck and Rosenthal mappings.

Findings

No new ideals for p-compact, weakly p-compact, and related classes.

New weighted holomorphic ideals arise for Grothendieck and Rosenthal mappings.

The procedure's applicability varies depending on the operator ideal considered.

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.