## 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

### Keywords:

Hardy space; unit disc; inner function; upper halfplane; moduli; arguments
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### References:

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