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Essential norms of composition operators between Bloch type spaces. (English) Zbl 1190.47028
Summary: For $ \alpha>0$, the $ \alpha$-Bloch space is the space of all analytic functions $ f$ on the unit disk $ D$ satisfying $$\Vert f\Vert _{B^{\alpha}}=\sup_{z\in D}\vert f^{\prime}(z)\vert(1-\vert z\vert^2)^{\alpha}<\infty.$$ Let $ \varphi$ be an analytic self-map of $ D$. We show that for $ 0<\alpha,\beta<\infty$, the essential norm of the composition operator $ C_{\varphi}$ mapping from $ B^{\alpha}$ to $ B^{\beta}$ can be given by the following formula: $$\Vert C_{\varphi}\Vert _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\Vert\varphi^n\Vert _{B^{\beta}}.$$

47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions
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