26. CMB 2001 (vol 44 pp. 270)
 Cheung, WaiShun; Li, ChiKwong

Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$.
The $c$numerical range of an $n \times n$ matrix $A$ is defined by
$$
W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\}
\text{ an orthonormal basis for } \IC^n \Bigr\},
$$
and the $c$numerical radius of $A$ is defined by $r_c (A) = \max
\{z : z \in W_c (A)\}$. We determine the structure of those linear
operators $\phi$ on algebras of block triangular matrices, satisfying
$$
W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or}
\quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.
$$
Keywords:linear operator, numerical range (radius), block triangular matrices Categories:15A04, 15A60, 47B49 

27. CMB 2000 (vol 43 pp. 448)
 Li, ChiKwong; Zaharia, Alexandru

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a
subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider
the {\it generalized matrix function} on $m\times m$ matrices $B =
(b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m
b_{j\sigma(j)}$ and the {\it generalized numerical range} of an
$n\times n$ complex matrix $A$ associated with $d^H$ defined by
$$
\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that }
X^*X = I_m\}.
$$
It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$.
We show that there exist normal matrices $A$ for which $\wmp(A)$ is
not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove
that a normal matrix $A $ with eigenvalues lying on a straight line
has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some
nonzero $\nu \in \IC$. These results extend those of Hu, Hurley
and Tam, who studied the special case when $2 \le m \le 3 \le n$
and $H = S_m$.
Keywords:convexity, generalized numerical range, matrices Category:15A60 

28. CMB 1998 (vol 41 pp. 178)
29. CMB 1998 (vol 41 pp. 105)