## Spectra of some composition operators.(English)Zbl 0814.47040

If $$\mathcal H$$ is a Hilbert space of holomorphic functions on the unit ball $$B_ N$$ in $${\mathbf C}^ N$$ and $$\phi$$ is a non-constant holomorphic map of the unit ball into itself, the composition operator $$C_ \varphi$$ is the operator on $$\mathcal H$$ defined by $$C_ \varphi f= f\circ \varphi$$. The authors give spectral properties for bounded composition operators on some weighted Hardy spaces under the condition that $$\varphi$$ is univalent and has a fixed point in the ball. When $$\mathcal H$$ is the usual Hardy space or a standard Bergman space on the unit disk, these properties show that the spectrum of the composition operator is the disk centered at 0 whose radius is the essential spectral radius of the operator.

### MSC:

 47B38 Linear operators on function spaces (general)
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