Grant: Discovery Grants Program - Individual

This proposal is in Operator Theory and is devoted to investigating invariant subspaces for finite-rank perturbations of linear operators acting on infinite dimensional Banach spaces. The Invariant Subspace Problem (ISP) is one of the most famous open problems in Operator Theory.  I plan continue a very successful investigation into the ISP for finite-rank perturbations that I initiated in a previous NSERC DG grant proposal. The original method I introduced proved to be very effective and produced some striking results. The long-term goal of the current proposal is to refine and enhance this method and expand it in several new and important directions, all part of the same overarching theme of my research program. I am summarizing below these directions; each has concrete short-term goals that are detailed in the Research Proposal.  A very important research direction is to examine the existence of reducing subspaces for finite rank perturbations of linear operators. A positive answer will substantially improve on an old theorem of Voiculescu, which showed there exist compact perturbations that have infinite dimensional and codimensional reducing subspaces. Another line of research deals with the existence of invariant subspaces for quasinilpotent (qn) operators, a class of operators for which ISP is still open. In a recent paper I proved a direct link between "perturbation" type questions and the ISP for qn operators. I showed that a qn operator has an invariant subspace if and only if there exist perturbations by two rank one operators that are scalar multiples of each other, which are also qn. The question I plan to examine is: For a bounded operator acting on a separable, infinite dimensional Banach space, can we always find a single rank one perturbation that is also qn? A positive solution to this question, in particular a concrete construction of a such perturbation, will provide a better understanding of the ISP for qn operators. Another very interesting line of investigation is to examine these questions in spaces with "few operators". A Banach space is said to be hereditarily indecomposable (HI) if no infinite dimensional subspace can be written as a direct sum of further infinite dimensional subspaces. It is known that HI spaces have "few operators", in the sense that every bounded operator on an HI space is a multiple of identity plus a strictly singular operator. In 2009 Argyros and Haydon constructed an HI space in which every operator is a multiple of identity plus compact, thus providing the first known example of a Banach space on which every operator has an invariant subspace. It is not known whether every HI space has this property. When it comes to the ISP, the only strictly singular operators that we need to examine are the ones that are also quasinilpotent. It is natural to examine the question presented in the second paragraph in the context of HI spaces, where every "interesting" operator is already quasinilpotent.