Analysis of an inequality concerning perturbation of self-adjoint operators

In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist(σ(A),σ(B)) \geq δ then there is some $c\sb{sa} < 2$ (independent of A and B) such that c\sb{sa}\vert\vert\vert AQ - QB\vert\vert\vert \geq δ\vert\vert\vert Q\vert\vert\vert for any Q and any unitary invariant norm $\vert\vert\vert \cdot\vert\vert\vert$. Sz.-Nagy subsequently noted that c\sb{sa} \leq π/2. It has not been known, however, if π/2 is sharp. In this dissertation we analyze $\vert\vert\vert AQ - QB\vert\vert\vert$ relative to $\vert\vert\vert Q\vert\vert\vert$ in certain special cases in order to sharpen this inequality. We define $c\sb{n}$ as the smallest constant such that c\sb{n}\Vert AQ - QB\Vert \geq δ\Vert Q\Vert when A, Q and B are $n \times n$ matrices and $\Vert \cdot\Vert$ is the usual norm. We then prove $c\sb2$ = $\sqrt{3/2}\approx$ 1.22474, $c\sb3$ = (8 + 5$\sqrt{10}$)/18 $\approx$ 1.32285, and $\lim\limits\sb{n \to \infty}$ $c\sb{n}$ = π/2 $\approx$ 1.57080. In particular, this proves π/2 is sharp in this operator inequality.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• McEachin, Raymond Vincent, Jr

• Mathematics

Statement dc:rights

• Copyright 1990 McEachin, Raymond Vincent, Jr

Language dc:language

eng

Identifier

AAI9114341
(UMI)AAI9114341

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/20056