A conditional proof of the ISP for quasinilpotent operators

TL;DR

This paper introduces a new conjecture and provides a conditional proof that all quasinilpotent operators on complex Hilbert spaces have nontrivial invariant subspaces, advancing the understanding of the invariant subspace problem.

Contribution

It presents a novel conjecture and a conditional proof for the invariant subspace existence for quasinilpotent operators, a key case in the ISP.

Findings

Conditional proof of invariant subspace existence for quasinilpotent operators

Introduction of a new conjecture supported by heuristic reasoning

Posing of an open problem with significant implications for the ISP

Abstract

The invariant subspace problem (ISP) is a well known unsolved problem in funtional analysis. While many partial results are known, the general case for complex, infinite dimensional separable Hilbert spaces is still open. It has been shown that the problem can be reduced to the case of operators which are norm limits of nilpotents. One of the most important subcases is the one of quasinilpotent operators, for which the problem has been extensively studied for many years. In this paper, we will introduce a new conjecture (supported by a heuristic argument), and we will prove conditionally that every quasinilpotent operator has a nontrivial invariant subspace. We will conclude by posing an open problem which would have deep implications regarding the ISP.