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 \(\mathbb U\).


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