Essential norms of composition operators between Bloch type spaces

Zhao, Ruhan

doi:10.1090/S0002-9939-10-10285-8

Essential norms of composition operators between Bloch type spaces
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Proc. Amer. Math. Soc. 138 (2010), 2537-2546 Request permission

Abstract:

For $\alpha >0$, the $\alpha$-Bloch space is the space of all analytic functions $f$ on the unit disk $D$ satisfying \[ \|f\|_{B^{\alpha }}=\sup _{z\in D}|f’(z)|(1-|z|^2)^{\alpha }<\infty . \] 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: \[ \|C_{\varphi }\|_e=\left (\frac {e}{2\alpha }\right )^{\alpha }\limsup _{n\to \infty } n^{\alpha -1}\|\varphi ^n\|_{B^{\beta }}. \]

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