# zbMATH — the first resource for mathematics

Essential norms and stability constants of weighted composition operators on $$C(X)$$. (English) Zbl 1060.47033
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 $$\varphi(\{x\in X: | u(x)| \geq r\})$$ $$(r>0)$$.

##### MSC:
 47B33 Linear composition operators 34K20 Stability theory of functional-differential equations
Full Text: