Invariant subspaces of idempotents on Hilbert spaces

TL;DR

This paper explores the invariant subspace problem for operators on Hilbert spaces, establishing equivalences involving quasinilpotent operators and idempotents, and providing geometric characterizations and classifications.

Contribution

It offers a new equivalence condition for invariant subspaces related to quasinilpotent operators and idempotents, along with geometric insights and classifications.

Findings

Equivalence between invariant subspaces of quasinilpotent operators and pairs of idempotents with quasinilpotent commutator

Geometric characterization of invariant subspaces of idempotents

Classification of essentially idempotent operators

Abstract

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.