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Composition operators on Banach spaces of formal power series. (English) Zbl 1150.47014

Let \((\beta_n)\) be a sequence of positive numbers and take \(1 \leq p < \infty\) and \(q\) such that \(p^{-1}+ q^{-1}=1\). Consider the space \(H^p(\beta)\) of all formal power series \(f(z)=\sum a_n z^n\) such that \(\sum | a_n |^p \beta_n^p < \infty\). Under the additional hypothesis that there exists a non-negative integer \(j\) such that \(\sum n^{qj} \beta_n^{-q} < \infty\), and using well-known techniques introduced by A. L. Shields, the authors prove that if a composition operator \(C_\phi\) is compact on \(H^p(\beta)\), then the modulus of the non-tangential limit of \(\phi^{(j+1)}\) is greater than one at every point of the unit circle. It is also shown that if \(C_\phi\) is Fredholm on \(H^p(\beta)\), then \(\phi\) must be an autormophism of the open unit disk.
Rewiever’s remark: It appears that the proofs cannot be applied for the case \(p=1\) and, at least, this case should have been treated separately; for instance, for \(p=1\), so that \(q =\infty\), what is the formulation of the the condition \(\sum n^{qj} \beta_n^{-q} < \infty\)?

MSC:

47B33 Linear composition operators
46B99 Normed linear spaces and Banach spaces; Banach lattices
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