Inner functions, Bloch spaces and symmetric measures. (English) Zbl 1085.46020

In the main theorem of the paper under review, the authors construct an inner function \(I\) which, in a sense, decreases hyperbolic distances as much as desired for \(| z| \to1\). Then this theorem is applied to prove several results on composition operators, Zygmund functions and the existence of certain singular measures. Concerning composition operators, one application is the following theorem: Given any sequence \(\{f_n\}\) of analytic functions in the unit disk \(\mathcal D\), there exists an inner function \(I\) such that \(f_n\circ I\in\mathcal B_0\) (the little Bloch space) for \(n=1,2,3,\dots\).


46E15 Banach spaces of continuous, differentiable or analytic functions
30D45 Normal functions of one complex variable, normal families
30D50 Blaschke products, etc. (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47B38 Linear operators on function spaces (general)
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