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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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