## Spectra of composition operators on the Bloch and Bergman spaces.(English)Zbl 1024.47009

Let $$U$$ be the unit disc in the complex plane and $$\varphi$$ be an analytic map from $$U$$ to itself. In [J. Am. Math. Soc. 10, 299-325 (1997; Zbl 0870.30018)], P. S. Bourdon and J. H. Shapiro give the essential spectral radius formula on the Hardy space $$H^p$$, for any analytic $$\varphi$$ and $$0<p<\infty$$. Using the Calderón theory of complex interpolation, the authors obtain an analogous result on the Bergman space $$A^p$$, namely that for $$1<p<\infty$$, $\bigl( r_{e,A^p}(C_\varphi) \bigr)^p = \bigl( r_{e,A^2}(C_\varphi) \bigr)^2.$ They also obtain the spectrum of the composition operator $$C_\varphi$$ on the Bergman space, Bloch space, BMOA and Hardy space for $$\varphi$$ univalent, not an automorphism, with fixed point in $$U$$.
Reviewer: Jinkee Lee (Pusan)

### MSC:

 47B33 Linear composition operators 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables 32A36 Bergman spaces of functions in several complex variables

### Keywords:

spectrum; composition operator; Bloch space; Bergman space

Zbl 0870.30018
Full Text:

### References:

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