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Continuous and compact embeddings between star-invariant subspaces. (English) Zbl 0969.30018
Havin, V. P. (ed.) et al., Complex analysis, operators, and related topics. The S. A. Vinogradov memorial volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 113, 65-76 (2000).
Author’s abstract: Given an inner function $$\theta$$ on the upper half-plane $$\mathbb{C}_+$$, let $$K^p_\theta\overset\text{def}=H^p\cap \theta \overline H^p$$ be the corresponding star-invariant subspace of the Hardy space $$H^p=H^p (\mathbb{C}_+)$$. It has been previoulsy shown by the author that the (continuous) embedding relation $$K^p_\theta \subset K^q_\theta$$ holds, for $$1<p< q<\infty$$, if and only if $$\theta'\in L^\infty (\mathbb{R})$$. In this paper, we first give an inextended version of the above result and then establish a compactness criterion for the embedding operator involved. Namely, we prove that the inclusion map $$id:K^p_\theta \hookrightarrow K^q_\theta$$ is compact if and only if $$\theta'\in C_0 (\mathbb{R})$$. Similar results are obtained on the embeddings $$K^p_\theta \subset C_0(\mathbb{R})$$ and $$K^p_\theta\subset \text{BMO}$$.
For the entire collection see [Zbl 0934.00031].

##### MSC:
 30D55 $$H^p$$-classes (MSC2000) 47B38 Linear operators on function spaces (general)
BMO