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Zbl 1190.47028
Zhao, Ruhan
(Zhao, Ru-han)
Essential norms of composition operators between Bloch type spaces.
(English)
[J] Proc. Am. Math. Soc. 138, No. 7, 2537-2546 (2010). ISSN 0002-9939; ISSN 1088-6826/e

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: $$\Vert C_{\varphi}\Vert _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\Vert\varphi^n\Vert _{B^{\beta}}.$$
MSC 2000:
*47B33 Composition operators
46E15 Banach spaces of functions defined by smoothness properties

Keywords: composition operators; essential norms; Bloch type spaces

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