Compact weighted composition operators on function algebras
Let \({\mathcal M}_ A\) denote the maximal ideal space of the uniform algebra A, let \({\mathcal M}^{\infty}_ A={\mathcal M}_ A\cup \{0\}\) and let {\^{\ }} denote the Gelfand transform. A continuous linear oprator T from A into A is called a weighted composition operator if there exists \(u\in A\) and a continuous mapping \(\phi\) from support(u) into \({\mathcal M}^{\infty}_ A\) such that \[ Tf(x)=u(x)\hat f(\phi (x))\quad if\quad x\in \sup port(u)\quad and\quad =0\quad if\quad x\not\in \sup port(u). \] In this article the author investigates conditions under which weighted composition operators are compact. Using Gleason parts and an analyticity criterion due to \textit{S. Ohno} and \textit{J. Wada} [Tokyo J. Math. 4, 105- 112 (1981; Zbl 0471.46035)] the author characterizes the compact weighted composition operators of \(H^{\infty}(D)\) and recovers known results for such operators on \({\mathcal C}(X)\) and the disc algebra.
scientific article; zbMATH DE number 4084409