Triangularization of a Jordan Algebra of Schatten Operators

TL;DR

This paper proves that certain Jordan algebras of Schatten operators are simultaneously triangularizable, demonstrating the existence of common invariant subspaces under specific algebraic conditions.

Contribution

It establishes that Jordan algebras of quasinilpotent Schatten operators containing a nonzero trace class operator have a common invariant subspace, leading to triangularization results.

Findings

Jordan algebra of compact quasinilpotent operators with a trace class operator has a common invariant subspace

Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable

Provides conditions for invariant subspace existence in operator algebras

Abstract

We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.