Soft ideals and arithmetic mean ideals

TL;DR

This paper explores the interaction between soft operations and arithmetic mean operations on operator ideals, providing characterizations of their interiors and covers, which are crucial for understanding traces and ideal structures.

Contribution

It explicitly analyzes the interplay between soft-interior, soft-cover, and arithmetic mean operations on operator ideals, extending the understanding of their structural properties.

Findings

Characterization of the am-interior of an ideal

Characterization of the am-infinity interior of an ideal

Analysis of the commutation relations between soft and arithmetic mean operations

Abstract

This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are "soft", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and arithmetic mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal.