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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