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Compact composition operators on the Bloch space. (English) Zbl 0826.47023
Summary: Necessary and sufficient conditions are given for a composition operator \(C_\phi f= f\circ \phi\) to be compact on the Bloch space \(\mathcal B\) and on the little Bloch space \({\mathcal B}_0\). Weakly compact composition operators on \({\mathcal B}_0\) are shown to be compact. If \(\phi\in {\mathcal B}_0\) is a conformal mapping of the unit disk \(\mathbb{D}\) into itself whose image \(\phi(\mathbb{D})\) approaches the unit circle \(\mathbb{T}\) only in a finite number of nontangential cusps, then \(C_\phi\) is compact on \({\mathcal B}_0\). On the other hand if there is a point of \(\mathbb{T}\cap \overline{\phi(\mathbb{D})}\) at which \(\phi(\mathbb{D})\) does not have a cusp, then \(C_\phi\) is not compact.

MSC:
47B38 Linear operators on function spaces (general)
47B07 Linear operators defined by compactness properties
30D55 \(H^p\)-classes (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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