The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.

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 $$ W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A. $$

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 $H = S_m$.

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