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An extension of a convexity theorem of the generalized numerical range associated with

Authors:	Tin-Yau Tam

Institution:	Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310

Abstract:	For any $C, A_1, A_2, A_3 \in {\frak {so}}(2n+1)$ , let 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 is always convex. When , 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 ${\frak u}(n)$ .

Keywords:	Numerical range convexity special orthogonal group weak majorization

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