Composition operators acting between Hardy spaces (Q5953396)

The author shows that for \(0<q<p<\infty\), the composition operator \(C_\varphi:H^p\to H^q, f\mapsto f\circ \varphi\), is compact if and only if \(|\varphi|<1\) almost everywhere on the unit circle. It is also shown that the difference of two composition operators \(C\varphi\) and \(C_\psi\) between \(H^p\) and \(H^q\) is compact if and only if both \(C_\varphi\) and \(C_\psi\) are compact. Reviewer's remark: the pages 390 and 391have to be interchanged.