Let \(\Omega\subset{\mathbb{C}}^ n\) be a domain and \(\Phi: \Omega\to \Omega\) a mapping. The operator \(T: f\to f\circ \Phi\) is called a composition operator.
The subject of composition operators represents a fertile arena for the interaction of operator theory, hard analysis, and geometry. Only a few dozen papers have been written in the field so far, and these have been primarily concerned with function spaces on homogeneous domains - mainly balls and polydiscs. I would like to see the theory of composition operators developed on, say, strongly pseudoconvex domains in \({\mathbb{C}}^ n.\) The opportunities to relate deep properties of Kähler geometry to deep properties of canonical operators seem manifest.
J. Shapiro is one of the foremost workers in the field of composition operators, and this paper represents a high point in the subject. He obtains a complete characterization of compact composition operators on \(H^ 2(D)\), \(D=\{z\in {\mathbb{C}}:| z| <1\}\), together with a number of interesting consequences for peak sets, essential norm of composition operators, etc.
I recommend this paper as a delightful introduction to an important topic that has not been sufficiently explored.