IN KOENIGS' FOOTSTEPS: DIAGONALIZATION OF COMPOSITION OPERATORS
Abstract
Let ϕ : D → D be a holomorphic map with a fixed point α ∈ D such that 0 ≤ |ϕ (α)| < 1. We show that the spectrum of the composition operator C ϕ on the Fréchet space Hol(D) is {0}∪{ϕ (α) n : n = 0, 1, · · · } and its essential spectrum is reduced to {0}. This contrasts the situation where a restriction of C ϕ to Banach spaces such as H 2 (D) is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schröder symbol on arbitrary Banach spaces of holomorphic functions.
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Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
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