Invariant subspaces of subgraded Lie algebras of compact operators

TL;DR

This paper investigates conditions under which subgraded Lie algebras of compact operators possess invariant subspaces, providing new structural insights and criteria for triangularizability.

Contribution

It establishes invariant subspace existence under quasinilpotence conditions and introduces new criteria for triangularizability of Lie algebras of compact operators.

Findings

Invariant subspaces exist under quasinilpotence conditions

New criteria for triangularizability of Lie algebras

Structural insights into subgraded Lie algebras of compact operators

Abstract

We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators.