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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 ${W}_{\psi ,\varphi }f=\psi \left(f\circ \varphi \right)$ acting on $BMOA$ and $VMOA$ of the unit disc. The results extend and unify those known for the cases $\varphi \left(z\right)=z$ and $\psi \left(z\right)=1$ corresponding to the multiplication operator ${M}_{\psi }$ [see S. Janson, Ark. Mat. 14, 189–196 (1976; Zbl 0341.43005) and D. A. Stegenga, Am. J. Math. 98, 573–589 (1976; Zbl 0335.47018)] and the composition operator ${C}_{\varphi }$ [see P. S. Bourdon, J. A. Cima and A. L. Matheson, Trans. Am. Math. Soc. 351, No. 6, 2183–2196 (1999; Zbl 0920.47029) and 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 $\alpha \left(\psi ,\varphi ,a\right)=|\psi \left(a\right)|\parallel {\sigma }_{\varphi \left(a\right)}\circ \varphi \circ {\sigma }_{a}{\parallel }_{{H}^{2}}$ and $\beta \left(\psi ,\varphi ,a\right)=\left(log\frac{2}{1-{|\varphi \left(a\right)|}^{2}}\right){\parallel \psi \circ {\sigma }_{a}-\psi \left(a\right)\parallel }_{{H}^{2}}$, where ${\sigma }_{a}$ stands for the Möbious transform mapping $\sigma \left(0\right)=a$, are bounded for $|a|<1$. 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