Arazy-type decomposition theorem for bounded linear operators and commutators on the trace class

TL;DR

This paper extends Arazy's decomposition theorem to general bounded operators on separable operator ideals, providing new tools for analyzing commutators and strictly singular operators in noncommutative settings.

Contribution

The authors generalize Arazy's decomposition theorem to a broader class of operators on separable operator ideals, enabling new characterizations of commutators and strictly singular operators.

Findings

Characterization of commutators on Schatten-von Neumann classes

Operators on  are commutators iff not of form I+K with  strictly singular

Extended decomposition theorems for noncommutative operator analysis

Abstract

The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal $\mathcal C_E$ of the algebra $\mathcal B(H)$ of all bounded linear operators on the separable infinite-dimensional Hilbert space $H$. In this paper, we extend and strengthen Arazy's decomposition theorem to the setting of general bounded linear operators on a separable (quasi-Banach) operator ideal $\mathcal C_E$ of $\mathcal B(H)$. Several applications are given to the study of $\mathcal C_E$-strictly singular operators, largest proper ideals in the algebra $\mathcal B(\mathcal C_E)$ of all bounded linear operators on $\mathcal C_E$ and complementably homogeneous Banach spaces among others. Our versions of decomposition theorems supply tools for a noncommutative generalization of deep commutator theorems for operators on $\ell_p$ and $L_p$, $1\le p <\infty $, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class $\mathcal C_p$, $1\le p<\infty $. For the crucial case, $p=1$, we establish that any operator $T\in\mathcal B(\mathcal C_1)$ is a commutator if and only if $T$ is not of the form $\lambda I+K$ for some $\lambda\neq 0$ and $\mathcal C_1$-strictly singular operator $K$.