Compressible Matrix Algebras and the Distance from Projections to Nilpotents

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.