A weighted endomorphism of an algebra is an endomorphism followed by a multiplier. In  and , H. Kamowitz characterized compact weighted endomorphisms of $C(X)$ and the disc algebra. In this note we define a weighted composition operator on a function algebra as a generalization of a weighted endomorphism, and characterize compact weighted composition operators on a function algebra satisfying a certain condition [Theorem 2]. This theorem not only includes Kamowitz's results as corollaries, but also has an application to compact weighted composition operators on the Hardy class $H^\infty(D)$.
"Compact Weighted Composition Operators on Function Algebras." Tokyo J. Math. 11 (1) 119 - 129, June 1988. https://doi.org/10.3836/tjm/1270134266