In Koenigs' footsteps: diagonalization of composition operators

Wolfgang Arendt (1) , Benjamin Célariès (2, 3, 4, 5) , Isabelle Chalendar (6)

1 Institut für Angewandte Analysis
2 ECL - École Centrale de Lyon
3 UCBL - Université Claude Bernard Lyon 1
4 ICJ - Institut Camille Jordan
5 EDPA - Équations aux dérivées partielles, analyse
6 UPEM - Université Paris-Est Marne-la-Vallée

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.

Domaines: Mathématiques [math]; Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]; Théorie spectrale [math.SP]

Wolfgang Arendt, Benjamin Célariès, Isabelle Chalendar. In Koenigs' footsteps: diagonalization of composition operators. Journal of Functional Analysis, 2020, 278 (2), pp.108313. ⟨10.1016/j.jfa.2019.108313⟩. ⟨hal-02065450v2⟩