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Compact composition operators on the Smirnov class. (English) Zbl 0954.47028

The authors show that a composition operator \[ G_{\varphi }f(z)=f(\varphi (z)),z\in D, \] where \(\varphi \) is a holomorphic mapping of the unit disk in \(\mathbb{C}\) is compact on the Smirnov class \(N^{+}\) if and only if it is compact on some (equivalently: every) Hardy space \(H^{p}(D)\) , \(0<p<\infty .\)

MSC:

47B33 Linear composition operators
47B07 Linear operators defined by compactness properties
47B38 Linear operators on function spaces (general)
30D55 \(H^p\)-classes (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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