Revisiting Operator $p$-Compact Mappings

TL;DR

This paper explores the geometric aspects of operator p-compact maps, examining their behavior in biduals and their relationships with other operator ideals, advancing the theoretical understanding of these mappings.

Contribution

It introduces a geometric perspective on operator p-compactness, including a quantitative approach and analysis in bidual spaces, linking it to other key operator ideals.

Findings

Quantitative notion of operator p-compactness for matrix sets

Analysis of operator p-compactness in bidual spaces

Connections established between p-compact maps and other operator ideals

Abstract

We continue our study of the mapping ideal of operator $p$-compact maps, previously introduced by the authors. Our approach embraces a more geometric perspective, delving into the interplay between operator $p$-compact mappings and matrix sets, specifically we provide a quantitative notion of operator $p$-compactness for the latter. In particular, we consider operator $p$-compactness in the bidual and its relation with this property in the original space. Also, we deepen our understanding of the connections between these mapping ideals and other significant ones (e.g., completely $p$-summing, completely $p$-nuclear).