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THE SPECTRAL SCALE AND THE $k$–NUMERICAL RANGE

Published online by Cambridge University Press:  31 July 2003

CHARLES A. AKEMANN
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106–3038, U.S.A. e-mail: [email protected]
JOEL ANDERSON
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802–6401, U.S.A. e-mail: [email protected]
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Abstract

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Suppose that $c$ is a linear operator acting on an $n$-dimensional complex Hilbert Space $H$, and let $\tau$ denote the normalized trace on $B(H)$. Set $b_1 = (c+c^*)/2$ and $b_2 = (c-c^*)/2i$, and write $B$ for the spectral scale of $\{b_1, b_2\}$ with respect to $\tau$. We show that $B$ contains full information about $W_k(c)$, the $k$-numerical range of $c$ for each $k = 1,\dots,n$. This is in addition to the matrix pencil information that has been described in previous papers. Thus both types of information are contained in the geometry of a single 3-dimensional compact, convex set. We then use spectral scales to prove a new fact about $W_k(c)$. We show in Theorem 3.4 that the point $\lambda$ is a singular point on the boundary of $W_k(c)$ if and only if $\lambda$ is an isolated extreme point of $W_k(c)$: i.e. it is the end point of two line segments on the boundary of $W_k(c)$. In this case $\lambda = (n/k)\tau(cz)$, where $z$ is a central projection in the algebra generated by $c$ and the identity. In addition we show how the general theory of the spectral scale may be used to derive some other known properties of the $k$-numerical range.

Keywords

Type
Research Article
Copyright
2003 Glasgow Mathematical Journal Trust