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Differentiation in star-invariant subspaces. I: Boundedness and compactness. (English) Zbl 1011.47005
Given two inner functions \(\theta_1,\theta_2\) on the upper half-plane \(\mathbb{C}_+\), let \(K^p(\theta_1, \theta_2)= \overline\theta_1 H^p\cap \theta_2 \overline{H^p}\), where \(H^p=H^p (\mathbb{C}_+)\) is the Hardy space, \(p\geq 1\). It is shown that the operator \({d\over dx}: K^p(\theta_1, \theta_2)\to L^p\) is bounded iff \(\theta_1', \theta_2'\in L^\infty (\mathbb{R})\); moreover, the norm of the operator is equivalent to \(\|\theta_1' \|_\infty+ \|\theta_2' \|_\infty\). In addition, the operator is compact iff \(\theta_1', \theta_2'\in C_0 (\mathbb{R})\).
This implies the following result. Let \({\mathcal R}_\Lambda^p\) be the closed subspace of \(L^p(\mathbb{R})\) generated by the rational functions \(\{(x-\lambda)^{-j}: 1\leq j\leq m(\lambda)\), \(\lambda\in \Lambda\}\), where \(\Lambda\) is a discrete subset of \(\mathbb{C}\setminus\mathbb{R}\). Then the operator \({d\over dx}: {\mathcal R}^p_\Lambda \to L^p\) is bounded iff \({\mathcal F}_\Lambda\in L^\infty (\mathbb{R}) \), and compact iff \(F_\Lambda\in C_0(\mathbb{R})\); here \[ {\mathcal F}_\Lambda (x)= \sum_{\lambda\in \Lambda}m (\lambda) {|\text{Im} \lambda|\over |x-\lambda |^2},\quad x\in\mathbb{R}. \] For part II, cf. ibid. 387–409 (2002; Zbl 1011.47006).

MSC:
47A15 Invariant subspaces of linear operators
30D50 Blaschke products, etc. (MSC2000)
30D55 \(H^p\)-classes (MSC2000)
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