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The Distance from a Rank $n-1$ Projection to the Nilpotent Operators on $\mathbb {C}^n$

Part of: Basic linear algebra General theory of linear operators

Published online by Cambridge University Press:  02 April 2020

Zachary Cramer
Zachary Cramer*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, OntarioN2L 3G1
*
e-mail: zcramer@uwaterloo.ca
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Abstract

Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. 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. Furthermore, we prove 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.

Keywords

projectionnilpotentmatrixoperator

MSC classification

Primary: 47A58: Operator approximation theory
Secondary: 15A99: Miscellaneous topics
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 54 - 74
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Research supported in part by NSERC (Canada).

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