Composition operators on weighted Hardy spaces. (English) Zbl 1085.47031

The paper is concerned with composition operators on weighted Hardy spaces of the type \(H^p(\beta)=\{f(z)=\sum a_nz^n: \| f\| ^p_\beta:=\sum| a_n| ^p\beta(n)^p<\infty\}\). Unfortunately, the paper contains some erroneous statements; in particular, not every composition operator \(f\mapsto f\circ \phi\) maps \(H^p(\beta)\) into itself. Also, the range of \(p\) has to exclude \(p=1\) in order that some statements and formulas make sense. No examples are given as to whether there really exist compact composition operators \(C_\phi\) on \(H^p(\beta) \) for the weights appearing in Theorem 1 and for which \(\phi\) actually has a finite angular derivative at some point.


47B33 Linear composition operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)