Operator $p$-compact mappings

TL;DR

This paper introduces operator p-compact mappings and right p-nuclear operators within operator spaces, establishing their properties, relationships, and connecting them to existing classes like Webster's operator $ extinfty$-compact mappings.

Contribution

It extends classical operator ideals to the operator space setting, defining new classes and relating them to known concepts, enriching the theory of operator space ideals.

Findings

Operator p-compact mappings and right p-nuclear operators are introduced.

The classes are related and given natural operator space structures.

Operator $ extinfty$-compact mappings coincide with Webster's class, enabling an operator space structure for it.

Abstract

We introduce the class of operator $p$-compact mappings and completely right $p$-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator $\infty$-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.