## Compressible Matrix Algebras and the Distance from Projections to Nilpotents

dc.contributor.author | Cramer, Zachary | |

dc.date.accessioned | 2019-11-15 13:38:43 (GMT) | |

dc.date.available | 2019-11-15 13:38:43 (GMT) | |

dc.date.issued | 2019-11-15 | |

dc.date.submitted | 2019-11-12 | |

dc.identifier.uri | http://hdl.handle.net/10012/15251 | |

dc.description.abstract | In this thesis we address two problems from the fields of operator algebras and operator theory. In our first problem, we seek to obtain a description of the unital subalgebras $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ with the property that $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Algebras with this property are said to be \textit{idempotent compressible}. Likewise, we wish to determine which unital subalgebras of $\mathbb{M}_n(\mathbb{C})$ satisfy the analogous property for projections (i.e., self-adjoint idempotents). Such algebras are said to be \textit{projection compressible}. We begin by constructing various examples of idempotent compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$ for each integer $n\geq 3$. Using a case-by-case analysis based on reduced block upper triangular forms, we prove that our list includes all unital projection compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ up to similarity and transposition. A similar examination indicates that the same phenomenon occurs in the case of unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$. We therefore demonstrate that the notions of projection compressibility and idempotent compressibility coincide for unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, and obtain a complete classification of the unital algebras admitting these properties up to similarity and transposition. In our second problem, we address the question of computing the distance from a non-zero projection to the set of nilpotent operators acting on $\mathbb{C}^n$. Building on MacDonald's results in the rank-one case, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$ is $\frac{1}{2}\sec\left(\frac{\pi}{\frac{n}{n-1}+2}\right)$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\in\mathbb{M}_n(\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Moreover, it is shown that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks. | en |

dc.language.iso | en | en |

dc.publisher | University of Waterloo | en |

dc.subject | matrix | en |

dc.subject | operator | en |

dc.subject | projection | en |

dc.subject | idempotent | en |

dc.subject | nilpotent | en |

dc.subject | compression | en |

dc.subject | projection compressible | en |

dc.subject | idempotent compressible | en |

dc.title | Compressible Matrix Algebras and the Distance from Projections to Nilpotents | en |

dc.type | Doctoral Thesis | en |

dc.pending | false | |

uws-etd.degree.department | Pure Mathematics | en |

uws-etd.degree.discipline | Pure Mathematics | en |

uws-etd.degree.grantor | University of Waterloo | en |

uws-etd.degree | Doctor of Philosophy | en |

uws.contributor.advisor | Marcoux, Laurent | |

uws.contributor.affiliation1 | Faculty of Mathematics | en |

uws.published.city | Waterloo | en |

uws.published.country | Canada | en |

uws.published.province | Ontario | en |

uws.typeOfResource | Text | en |

uws.peerReviewStatus | Unreviewed | en |

uws.scholarLevel | Graduate | en |