Weighted composition operators from Zygmund spaces into Bloch spaces.(English)Zbl 1215.47022

Let $$\mathbb{D}=\{ z\in \mathbb{C}:| z| <1\}$$ be the open unit disk in the complex plane $$\mathbb{C}$$ and $$H(\mathbb{D})$$ the class of all analytic functions on $$\mathbb{D}$$, and let $$\varphi$$ be an analytic self-map, that is, $$\varphi :\mathbb{D}\rightarrow \mathbb{D}$$. Then $$\varphi$$ induces the composition operator $$C_{\varphi }$$ on $$H(\mathbb{D}),$$ defined by $$(C_{\varphi })(z)=f(\varphi (z))$$ for $$z\in \mathbb{D}$$ and $$f\in H(\mathbb{D})$$. Let $$u$$ be a fixed analytic function on $$\mathbb{D}$$. The weighted composition operator $$uC_{\varphi }$$ is defined on $$H(\mathbb{D} )$$ by $$((uC_{\varphi })f)(z)=u(z)f(\varphi (z)),$$ $$z\in \mathbb{D},$$ $$f\in H( \mathbb{D}).$$
Let $$\mathcal{Z}$$ denote the space of all $$f\in H(\mathbb{D})\cap C( \overline{\mathbb{D}})$$ such that
$\| f\| _{\mathcal{Z}}=\sup \frac{| f(e^{i(\theta+h))}+f(e^{i(\theta -h)})-2f(e^{i\theta })| }{h}<\infty,$
where the supremum is taken over all $$e^{i\theta }\in \partial \mathbb{D}$$ and $$h>0$$. The little Zygmund space $$\mathcal{Z}_{0}$$ is defined by $$f\in \mathcal{Z}_{0}$$ if and only if $$\lim_{| z|\rightarrow 1}(1-| z|)| f''(z)|=0$$. An analytic function $$f$$ is said to belong to the Bloch space $$B$$ if $$B(f)=\sup_{z\in \mathbb{D}}(1-| z|^{2})| f'(z)| <\infty$$. Finally, the closed subspace $$B_{0}$$ (the little Bloch space) of $$B$$ consists of the functions $$f\in B$$ satisfying $$\lim_{| z| \rightarrow 1}(1-| z| ^{2})| f'(z)| =0$$. Let $$X_{1}$$ and $$X_{2}$$ be the Banach spaces. For a given operator $$A:X_{1}\rightarrow X_{2}$$, the problem of investigating its boundedness and compactness is a natural and important problem of operator theory.
In this paper, the authors study the boundedness and compactness of weighted composition operators from Zygmund space into Bloch space and little Bloch space, based mainly on the classical methods of function theory.

MSC:

 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 30H05 Spaces of bounded analytic functions of one complex variable
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References:

 [1] Boe, B.; Nikolau, A., Interpolation by functions in the Bloch space, J. anal. math., 94, 171-194, (2004) · Zbl 1094.30042 [2] Choe, B.; Koo, H.; Smith, W., Composition operators on small spaces, Integr. equat. oper. th., 56, 357-380, (2006) · Zbl 1114.47028 [3] D. Clahane, S. Stević, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl., vol. 2006, 2006, 11 p., Article ID 61018. · Zbl 1131.47018 [4] Cowen, C.C.; MacCluer, B.D., Composition operators on spaces of analytic functions, Studies in advanced mathematics, (1995), CRC Press Boca Raton · Zbl 0873.47017 [5] Duren, P.L., Theory of $$H^p$$ spaces, (1970), Academic press New York · Zbl 0215.20203 [6] X. Fu, X. Zhu, Weighted composition operators on some weighted spaces in the unit ball, Abstr. Appl. Anal., vol. 2008, 2008, 7 p., Article ID 605807. · Zbl 1160.47024 [7] S. Li, S. Stević, Volterra type operators on Zygmund space, J. Inequal. Appl., vol. 2007, 2007, 10 p., Article ID 32124. [8] Li, S.; Stević, S., Weighted composition operators from $$\alpha$$-Bloch space to $$H^\infty$$ on the polydisk, Numer. funct. anal. optimiz., 28, 7, 911-925, (2007) · Zbl 1130.47015 [9] Li, S.; Stević, S., Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian acad. sci. math. sci., 117, 3, 371-385, (2007) · Zbl 1130.47016 [10] S. Li, S. Stević, Weighted composition operators from $$H^\infty$$ to the Bloch space on the polydisc, Abstr. Appl. Anal., vol. 2007, 2007, 12 p., Article ID 48478. · Zbl 1152.47016 [11] S. Li, S. Stević, Weighted composition operators between $$H^\infty$$ and $$\alpha$$-Bloch spaces in the unit ball, Taiwan. J. Math., in press. [12] Lou, Z., Composition operators on Bloch type spaces, Analysis (Munich), 23, 1, 81-95, (2003) · Zbl 1058.47024 [13] Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. am. math. soc., 347, 7, 679-2687, (1995) · Zbl 0826.47023 [14] Ohno, S., Weighted composition operators between $$H^\infty$$ and the Bloch space, Taiwanese J. math., 5, 3, 555-563, (2001) · Zbl 0997.47025 [15] Ohno, S.; Stroethoff, K.; Zhao, R., Weighted composition operators between Bloch-type spaces, Rocky mountain J. math., 33, 1, 191-215, (2003) · Zbl 1042.47018 [16] Stević, S., Composition operators between $$H^\infty$$ and the $$\alpha$$-Bloch spaces on the polydisc, Z. anal. anwend., 25, 4, 457-466, (2006) · Zbl 1118.47015 [17] Stević, S., Norm of weighted composition operators from Bloch space to $$H_\mu^\infty$$ on the unit ball, Ars. combin., 88, 125-127, (2008) · Zbl 1224.30195 [18] Tamrazov, P.M., Contour and solid structure properties of holomorphic functions of a complex variable, Russ. math. surveys, 28, 141-173, (1973) · Zbl 0273.30036 [19] K. Zhu, Operator Theory in Function Spaces, Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York and Basel, 1990. · Zbl 0706.47019 [20] Zhu, X., Weighted composition operators between $$H^\infty$$ and Bergman type spaces, Commun. Korean math. soc., 21, 4, 719-727, (2006) · Zbl 1160.47028 [21] Zhu, X., Generalized weighted composition operators from Bloch-type spaces to weighted Bergman spaces, Indian J. math., 49, 2, 139-149, (2007) · Zbl 1130.47017
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