## 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 $\| f\| _{B^{\alpha}}=\sup_{z\in D}| f^{\prime}(z)|(1-| z|^2)^{\alpha}<\infty.$ 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: $\| C_{\varphi}\| _e=\left(\frac{e}{2\alpha}\right)^{\alpha}\limsup_{n\to\infty} n^{\alpha-1}\|\varphi^n\| _{B^{\beta}}.$

### MSC:

 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions

### Keywords:

composition operators; essential norms; Bloch type spaces
Full Text:

### References:

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