## Publication Date

2016

## Document Type

Dissertation/Thesis

## First Advisor

Sirotkin, Gleb

## Degree Name

Ph.D. (Doctor of Philosophy)

## Legacy Department

Department of Mathematical Sciences

## LCSH

Banach spaces; Linear operators; Invariant subspaces

## Abstract

Let X be a complex, infinite-dimensional Banach space and T : X → X a continuous linear operator. A closed subspace Y of X such that Y and X/Y both have infinite dimension is called a halfspace. We say that Y is almost-invariant under T if there is a finite-rank linear operator F : X → X such that (T + F)Y ⊆ Y; in this case, the minimum possible rank of F is called the defect. If (T +K)Y ⊆ Y for some compact linear operator K : X → X then we say that Y is essentially-invariant under T. We show that if T is compact, strictly singular, weakly compact, quasinilpotent, or has at most countably many eigenvalues, then it admits an almost-invariant halfspace (AIHS) of defect [precedes or equal to] 1. We also improve an existing result by showing that T* always admits an AIHS of defect [precedes or equal to] 1, with no assumptions on T. Although it is not currently known whether every operator acting on a complex, infinite-dimensional Banach space admits an AIHS, we do nevertheless show that every such operator admits an essentially-invariant halfspace. Some additional results on AIHS's and their connection to invariant subspaces are also obtained.

## Recommended Citation

Wallis, Ben, "The almost-invariant halfspace problem" (2016). *Graduate Research Theses & Dissertations*. 5297.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/5297

## Extent

59 pages

## Language

eng

## Publisher

Northern Illinois University

## Rights Statement

In Copyright

## Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

## Media Type

Text

## Comments

Advisors: Gleb Sirotkin.||Committee members: Ilya Krishtal; Anders Linner; Gleb Sirotkin; Zhuan Ye.||Includes bibliographical references.