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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].

30D55 \(H^p\)-classes (MSC2000)
47B38 Linear operators on function spaces (general)