Spectral conditions on Lie and Jordan algebras of compact operators

TL;DR

This paper explores spectral properties of bounded operators in Lie and Jordan algebras of compact operators, establishing conditions for invariant subspaces and triangularizability.

Contribution

It introduces new spectral conditions that guarantee invariant subspaces and triangularizability in Lie and Jordan algebras of compact operators.

Findings

Algebras have nontrivial invariant subspaces under spectral additivity conditions.

Certain spectral conditions imply the algebra is triangularizable.

Results apply to operators with sublinear or submultiplicative spectra.

Abstract

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.