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An extension of a convexity theorem
of the generalized numerical range
associated with $SO(2n+1)$

Author: Tin-Yau Tam
Journal: Proc. Amer. Math. Soc. 127 (1999), 35-44
MSC (1991): Primary 15A60, 22E15
MathSciNet review: 1473680
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Abstract: For any $C, A_1, A_2, A_3 \in {\frak {so}}(2n+1)$, let $W_C(A_1, A_2, A_3)$ be the following subset of ${\mathbb R}^3$:

\begin{displaymath}\{(\operatorname{tr}CO^TA_1O, \operatorname{tr}CO^TA_2O, \operatorname{tr}CO^TA_3O): O\in SO(2n+1)\}. \end{displaymath}

We show that if $n\ge 2$, then $W_C(A_1, A_2, A_3)$ is always convex. When $n = 1$, it is an ellipsoid, probably degenerate. The convexity result is best possible in the sense that if we have $W_C(A_1, \dots, A_p)$ defined similarly, then there are examples which fail to be convex when $p \ge 4$ and $n\ge 1$.

The set is also symmetric about the origin for all $n\ge 1$, and contains the origin when $n \ge 2$. Equivalent statements of this result are given. The convexity result for ${\frak {so}}(2n+1)$ is similar to Au-Yeung and Tsing's extension of Westwick's convexity result for ${\frak u}(n)$.

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Tin-Yau Tam
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310

Keywords: Numerical range, convexity, special orthogonal group, weak majorization
Received by editor(s): November 26, 1996
Received by editor(s) in revised form: May 9, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society