## Hardy spaces that support no compact composition operators.(English)Zbl 1041.46019

Let $$G$$ be a simply connected domain which is properly contained in $$\mathbb C$$ and let $$\tau$$ be a Riemann map of the open unit disc $$\mathbb U$$ onto $$G$$. For $$0<p<\infty$$, the {Hardy space} $$\mathcal H^p(G)$$ consists of all analytic functions $$F:G\to\mathbb C$$ such that the integrals of $$|F|^p$$ over the curves $$\tau(|z|=r)$$, $$0<r<1$$, are bounded. The main theorem states that $$\mathcal H^p(G)$$ supports compact composition operators if and only if $$\partial G$$ has finite one-dimensional Hausdorff measure. Actually, this holds with ‘compact’ replaced by Riesz. Related results are obtained for Bergman spaces on $$G$$. There is also a characterization of domains $$G$$ for which every composition operator $$\mathcal H^p(G)$$ is bounded: for this it is necessary and sufficient that $$\tau'$$ and $$1/\tau'$$ are both bounded on $$\mathbb U$$.

### MSC:

 46E15 Banach spaces of continuous, differentiable or analytic functions 47B33 Linear composition operators
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### References:

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