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Spectra of composition operators on algebras of analytic functions on Banach spaces. (English) Zbl 1181.47020

The authors study composition operators acting on the uniform algebra of bounded analytic functions on an open unit ball in a complex Banach space. The generating symbols are assumed to have fixed points in the ball. The problem is to relate various properties of a composition operator to corresponding properties of another linear operator, namely, the Fréchet derivative of the symbol at its fixed point. Especially the spectra, essential spectra, and the property to be a Riesz operator are considered. This generalizes earlier results on compact composition operators. Among the tools there are lower triangular representations of composition operators corresponding to the Taylor series expansion at a fixed point of the symbol, interpolating sequences, and Julia-type estimates. In particular, a characterization of composition operators having essential spectral radius strictly less than 1 is given in terms of the behaviour of the iterates of the symbol.

MSC:

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