Differences of weighted composition operators on the disk algebra (Q961977)

Let \(\mathbb D\) be the open unit disk in the complex plane and \(A= A(\mathbb D)\) be the disk algebra of all continuous functions on \(\overline{\mathbb D}\), the closure of \(\mathbb D\), that are analytic on \(\mathbb D\). Denote by \(H^\infty\) the set of all bounded analytic functions on \(\mathbb D\). For \(u, \varphi \in A\) with \(\|\varphi\|_\infty \leq 1\), the weighted composition operator \(u C_\varphi\) defined on \(A\) is given by \(u C_\varphi f= u \cdot (f \circ \varphi)\) for \(f \in A\). In this paper, the authors give necessary and sufficient conditions for the difference \(u C_\varphi- v C_\psi\) of two weighted composition operators on \(A\) to be compact, weakly compact and completely continuous. Indeed, they show that they are all equivalent. The same result on the setting of \(H^\infty\) has been given in [\textit{T. Hosokawa, K. Izuchi} and \textit{S. Ohno}, Integral Equations Oper. Theory 53, No. 4, 509--526 (2005; Zbl 1098.47025)]. In the last section, they study the difference of such operators acting from \(H^\infty\) to \(A\), for which similar equivalent conditions are given.