1. CMB Online first
2. CMB 2008 (vol 51 pp. 86)
3. CMB 2003 (vol 46 pp. 216)
 Li, ChiKwong; Rodman, Leiba; Šemrl, Peter

Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range
Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of
bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH
\to \HH$ leaving invariant various properties such as invertibility, positive
definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed
{\it a priori\/} continuous. It is shown that under an appropriate surjective
or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto
\xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, 1\}$,
where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis.
Examples are given to show that the surjective or injective assumption cannot
be relaxed. The results are extended to complex linear maps on the algebra of
bounded linear operators on $H$. Similar results are proved for the (real)
linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$
is a scalar and $K$ is compact.
Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical range Categories:47B15, 47B49 

4. 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 

5. 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 
