Let \(X\) be a compact Hausdorff space and let \(C(X)\) denote the Banach space of all continuous functions on \(X\) with the supremum norm. The authors consider the weighted composition operator \(uC_{\varphi}\) on \(C(X)\) defined by \[ (uC_{\varphi}f)(x)=u(x)f(\varphi (x)) \qquad (x \in X) \] for all \(f \in C(X)\), where \(u\) is a fixed function in \(C(X)\) and \(\varphi\) is a selfmap of \(X\) which is continuous on the support \(S(u)\) of \(u\).
H. Kamowitz [Proc. Am. Math. Soc. 83, 517–521 (1981; Zbl 0509.47026)] characterized when \(uC_{\varphi}\) is compact. The authors develop Kamowitz’s result. They determine the essential norm of \(uC_{\varphi}\), that is, \(\| uC_{\varphi}\| _e =\inf\{r>0: \varphi(\{x\in X: | u(x)| \geq r\})\text{ is finite}\}\) (Theorem 1). Then, the authors give an equivalence proposition on the the Hyers-Ulam stability of a bounded linear operator between Banach spaces. With the aid of this proposition, they also characterize the Hyers-Ulam stability of \(uC_{\varphi}\) and determine the stability constant, in terms of the set \(\varphi(\{x\in X: | u(x)| \geq r\})\) \((r>0)\).