S-numbers of elementary operators on C*-algebras

TL;DR

This paper investigates the behavior of s-numbers of elementary operators on C*-algebras, establishing conditions under which their approximation sequences belong to stable Calkin spaces, and providing a quantitative extension of Ylinen's result.

Contribution

It introduces new conditions linking s-numbers of elementary operators to stable Calkin spaces and extends Ylinen's results with quantitative bounds.

Findings

Sequences of s-numbers of tensor products belong to stable Calkin spaces under certain conditions.

The s-number sequences of sums of elementary operators are in a stable Calkin space if their components are.

A converse characterization is provided when the ideal of compact elements has finite spectrum.

Abstract

We study the s-numbers of elementary operators acting on C*-algebras. The main results are the following: If $\tau$ is any tensor norm and $a,b\in B(H)$ are such that the sequences $s(a),s(b)$ of their singular numbers belong to a stable Calkin space $J$ then the sequence of approximation numbers of $a\otimes_{\tau} b$ belongs to $J$. If $A$ is a C*-algebra, $J$ is a stable Calkin space, $s$ is an s-number function, and $a_i, b_i \in A,$ $i=1,...,m$ are such that $s(\pi(a_i)), s(\pi(b_i)) \in J$, $i=1,...,m$ for some faithful representation $\pi$ of $A$ then $s(\sum_{i=1}^{m} M_{a_i,b_i})\in J$. The converse implication holds if and only if the ideal of compact elements of $A$ has finite spectrum. We also prove a quantitative version of a result of Ylinen.