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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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##### References:
  Ahern, P.R.; Clark, D.N., Radial limits and invariant subspaces, Amer. J. math., 92, 332-342, (1970) · Zbl 0197.39202  Ahern, P.R.; Clark, D.N., On inner functions with Hp-derivative, Michigan math. J., 21, 115-127, (1974) · Zbl 0277.30027  A. D. Baranov, Differentiation in de Branges spaces and embedding theorems, in Nonlinear Equations and Mathematical Analysis, J. Math. Sci. (New York)1012000, 2881-2913.  Calderón, A.P., Commutators of singular integral operators, Proc. natl. acad. sci. USA, 53, 1092-1099, (1965) · Zbl 0151.16901  Calderón, A.P., Cauchy integrals on Lipschitz curves and related operators, Proc. natl. acad. sci. USA, 74, 1324-1327, (1977) · Zbl 0373.44003  Cima, J.A.; Ross, W.T., The backward shift on the Hardy space, (2000), Amer. Math. Soc Providence · Zbl 0952.47029  Cohn, W.S., Radial limits and star invariant subspaces of bounded Mean oscillation, Amer. J. math., 108, 719-749, (1986) · Zbl 0607.30034  Douglas, R.G.; Sarason, D.E., A class of Toeplitz operators, Indiana univ. math. J., 20, 891-895, (1971) · Zbl 0199.19103  Douglas, R.G.; Shapiro, H.S.; Shields, A.L., Cyclic vectors and invariant subspaces for the backward shift operator, Ann. inst. Fourier (Grenoble), 20, 37-76, (1970) · Zbl 0186.45302  Dyakonov, K.M., Entire functions of exponential type and model subspaces in Hp, Zap. nauchn. sem. leningrad otdel. mat. inst. Steklov. (LOMI), 190, 81-100, (1991) · Zbl 0788.30024  Dyakonov, K.M., Smooth functions in the range of a Hankel operator, Indiana univ. math. J., 43, 805-838, (1994) · Zbl 0821.30026  Dyakonov, K.M., Continuous and compact embeddings between star-invariant subspaces, Oper. theory adv. appl., 113, 65-76, (2000) · Zbl 0969.30018  Garnett, J.B., Bounded analytic functions, (1981), Academic Press New York · Zbl 0469.30024  Havin, V.; Jöricke, B., The uncertainty principle in harmonic analysis, (1994), Springer-Verlag Berlin · Zbl 0827.42001  Koosis, P., Interior compact spaces of functions on a half-line, Comm. pure appl. math., 10, 583-615, (1957) · Zbl 0080.31801  Nikol’skiĭ, N.K., Treatise on the shift operator, (1986), Springer-Verlag Berlin  Stein, E.M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton Univ. Press Princeton · Zbl 0821.42001  Yosida, K., Functional analysis, (1965), Springer-Verlag Berlin · Zbl 0126.11504
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