Abstract. One of the natural problems on weighted composition operators $uC_\varphi $ is to determine the condition of $u$ and $\varphi $ for an operator $uC_\varphi $ to map bounded sets into sequentially compact sets, that is, for $uC_\varphi $ to become compact. In this paper, we shall characterize compact weighted composition operators on the domain of a closed $*$-derivation in $C(K)$ which is a generalization of spaces of differentiable functions.

AMS Subject Classification (1991): 47B38, 46E15, 46L57

Received June 29, 1994, and in revised form October 24, 1994. (Registered under 5701/2009.)