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26. CMB 2001 (vol 44 pp. 270)

Cheung, Wai-Shun; Li, Chi-Kwong
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, Chi-Kwong; 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)

Krupnik, Ilya; Lancaster, Peter
Minimal pencil realizations of rational matrix functions with symmetries
A theory of minimal realizations of rational matrix functions $W(\lambda)$ in the ``pencil'' form $W(\lambda)=C(\lambda A_1-A_2)^{-1}B$ is developed. In particular, properties of the pencil $\lambda A_1-A_2$ are discussed when $W(\lambda)$ is hermitian on the real line, and when $W(\lambda)$ is hermitian on the unit circle.

Categories:93Bxx, 15A23

29. CMB 1998 (vol 41 pp. 105)

So, Wasin
An explicit criterion for the convexity of quaternionic numerical range
Quaternionic numerical range is not always a convex set. In this note, an explicit criterion is given for the convexity of quaternionic numerical range.

Categories:15A33, 15A60
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