Königs eigenfunction for composition operators on Bloch and H∞ type spaces

Abstract

To Richard Aron on the occasion of his retirement, with very much appreciation Keywords: Composition operator Königs eigenfunctionWe discuss when the Königs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H ∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H ∞ for unbounded Königs eigenfunction are obtained.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

“…In [54,98] it was investigated how the essential spectral radius of C ϕ on both H ∞ v (D) and H 0 v (D) determines whether the Koenigs eigenfunction σ of ϕ belongs to H ∞ v (D) and H 0 v respectively. Let ϕ be an analytic self map on D which is not an automorphism such that ϕ(0) = 0 and 0 < |ϕ (0)| < 1.…”

“…characterization does not hold for more general radial weights. On the other hand, by [98,Theorem 3.1], σ ∈ H ∞ if and only if r e,H ∞ (C ϕ ) = 0.…”

“…Proposition 59 [98] Let ϕ be an analytic self map on D which is not an automorphism such that ϕ(0) = 0 and 0 < |ϕ (0)| < 1. Let v be a radial weight on D. Then (i) The Koenigs' eigenfunction σ belongs to H ∞ v (D) if and only if the sequence σ n := ϕ n /ϕ (0) n , n ∈ N, is bounded in H ∞ v (D).…”