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Weighted composition operators on BMOA. (English) Zbl 1163.47018
The author completely characterizes the boundedness and compactness of the weighted composition operator $W_{\psi,\varphi}f=\psi (f\circ \varphi)$ acting on $BMOA$ and $VMOA$ of the unit disc. The results extend and unify those known for the cases $\varphi(z)=z$ and $\psi(z)=1$ corresponding to the multiplication operator $M_\psi$ [see {\it S. Janson}, Ark. Mat. 14, 189--196 (1976; Zbl 0341.43005) and {\it D. A.\thinspace Stegenga}, Am. J. Math. 98, 573--589 (1976; Zbl 0335.47018)] and the composition operator $C_\varphi$ [see {\it P. S.\thinspace Bourdon, J. A.\thinspace Cima} and {\it A. L.\thinspace Matheson}, Trans. Am. Math. Soc. 351, No. 6, 2183--2196 (1999; Zbl 0920.47029) and {\it W. Smith}, Proc. Am. Math. Soc. 127, No. 9, 2715--2725 (1999; Zbl 0921.47025)]. The boundedness of $W_{\psi,\varphi}$ can be described by the facts that the two quantities $\alpha(\psi,\varphi,a)=\vert \psi(a)\vert \Vert \sigma_{\varphi(a)}\circ \varphi\circ \sigma_a\Vert _{H^2}$ and $\beta(\psi,\varphi,a)= (\log\frac{2}{1-\vert \varphi(a)\vert ^2})\Vert \psi\circ \sigma_a-\psi(a)\Vert _{H^2}$, where $\sigma_a$ stands for the Möbious transform mapping $\sigma(0)=a$, are bounded for $\vert a\vert <1$. The proof is based on a weighted version of the Littlewood subordination principle. The author also studies the case $VMOA$ and provides an asymptotic estimate for the essential norm of $W_{\psi,\varphi}$, which seems to be new even in the case of multiplication and composition operators

##### MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general) 30H05 Bounded analytic functions 30D50 Blaschke products, etc. (MSC2000) 46E15 Banach spaces of continuous, differentiable or analytic functions
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##### References:
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