Analysis of an inequality concerning perturbation of self-adjoint operators

McEachin, Raymond Vincent, Jr

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https://hdl.handle.net/2142/20056

Description

Title

Analysis of an inequality concerning perturbation of self-adjoint operators

Author(s)

McEachin, Raymond Vincent, Jr

Issue Date

1990

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

In 1983, Bhatia, Davis and McIntosh proved that if A and B are self-adjoint with dist($\sigma(A),\sigma(B)) \geq \delta$ then there is some $c\sb{sa} < 2$ (independent of A and B) such that $c\sb{sa}\vert\vert\vert AQ - QB\vert\vert\vert \geq \delta\vert\vert\vert Q\vert\vert\vert$ for any Q and any unitary invariant norm $\vert\vert\vert \cdot\vert\vert\vert$. Sz.-Nagy subsequently noted that $c\sb{sa} \leq \pi$/2. It has not been known, however, if $\pi/2$ 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 $c\sb{n}\Vert AQ - QB\Vert \geq \delta\Vert Q\Vert$ 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 $\lim\limits\sb{n \to \infty}$ $c\sb{n}$ = $\pi$/2 $\approx$ 1.57080. In particular, this proves $\pi$/2 is sharp in this operator inequality.

To prove the existence of $c\sb{sa}$ Bhatia, Davis and McIntosh first note that when A and B satisfy the given hypotheses the map ${\cal T}:Q \mapsto AQ - QB$ is invertible. Then they construct ${\cal T}\sp{-1}$ as a Fourier transform and show $\Vert {\cal T}\sp{-1}\Vert \leq 2/\delta$ by standard methods. Since we have $\Vert {\cal T}\sp{-1}\Vert \leq k$ for some k if and only if $k\vert\vert\vert AQ - QB\vert\vert\vert \geq \vert\vert\vert Q\vert\vert\vert,$ this proves $c\sb{sa} < 2.$ In 1987 Sz.-Nagy, referring to his earlier work, noted that if $f \in L\sb1$ and $\ f(s) = s\sp{-1}$ whenever $\vert s\vert$ $\geq$ 1 then $\Vert f\Vert\sb1$ $\geq$ $\pi$/2, and $\pi$/2 is sharp. As a result of the original construction of ${\cal T}\sp{-1}$, it follows immediately that $c\sb{sa} \leq \pi$/2.

For the cases when n = 2 or 3 we start with an extra assumption concerning the configuration of $\sigma(A)$ and $\sigma(B).$ Then we modify the Fourier transform argument to yield an upper bound on $\Vert {\cal T}\sp{-1}\Vert$. This bound is actually the correct value for $c\sb{n}$ when n is 2 or 3, but proving it requires more work. To justify our extra assumption we formulate the problem in terms of the Schur product of two matrices, so that for each A and B there is a matrix T for which ${\cal T}\sp{-1}X$ = $T \circ X$. Then we obtain estimates on $\Vert {\cal T}\sp{-1}\Vert$ from standard estimates on the Schur multiplier norm of a matrix T. From the specific information we obtain in our special case, combined with a basic inequality due to Ando, Horn and Johnson, we see that our upper bounds are correct no matter how $\sigma(A)$ and $\sigma(B)$ are configured. To supply lower bounds we give examples which are best possible. The ideas we've developed can then be extended to evaluate $c\sb{sa}$. We give $n \times n$ matrices $A\sb{n}$, $Q\sb{n}$ and $B\sb{n}$ such that $\delta\Vert Q\sb{n}\Vert/\Vert A\sb{n}Q\sb{n} - Q\sb{n}B\sb{n}\Vert \to \pi/2$ as $n \to \infty$. This proves that $\pi/2$ is sharp in the $AQ - QB$ inequality.

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