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Moduli and arguments of analytic functions from subspaces in \(H^ p\) that are invariant for the backward shift operator. (English. Russian original) Zbl 0728.32003

Sib. Math. J. 31, No. 6, 926-939 (1990); translation from Sib. Mat. Zh. 31, No. 6(184), 64-79 (1990).
Let \(K^ p_{\theta}({\mathbb{D}}):=H^ p({\mathbb{D}})\cap \theta \overline{H^ p_ 0({\mathbb{D}})}\), where \(H^ p({\mathbb{D}})\) is the p-th Hardy space in the unit disc \({\mathbb{D}}\), \(\theta\) is an inner function on \({\mathbb{D}}\) and \(H^ p_ 0({\mathbb{D}}):=\{f\in H^ p({\mathbb{D}}):\) \(f(0)=0\}\). Similarly, let \(K^ p_{\theta}({\mathbb{C}}_+):=H^ p({\mathbb{C}}_+)\cap \theta \overline{H^ p({\mathbb{C}}_+)}\), where \({\mathbb{C}}_+\) denotes the upper halfplane.
The author presents several properties of the spaces \(K^ p_{\theta}\), in particular, he characterizes
- the class of all functions from \(K^ p_{\theta}\) which are uniquely determined by their moduli,
- the class of all functions from \(K^ 1_{\theta}\) which are uniquely determined by their arguments,
-the class of those inner functions \(\theta\) for which there is a continuous embedding of \(K^ p_{\theta}\) into \(K^ q_{\theta}\) for \(p<q\).

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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