Spectra of compact composition operators

Abstract

Let ϕ \phi be holomorphic and map the open unit disk into itself, and let C ϕ : f → f ∘ ϕ {C_\phi }:f \to f \circ \phi be the composition operator on H 2 {H^2} generated by ϕ \phi . If C ϕ {C_\phi } is a compact operator then ( 1 ) ϕ ( z 0 ) = z 0 (1)\phi ({z_0}) = {z_0} for some z 0 ϵ D {z_0} \epsilon D ; ( 2 ) σ ( C ϕ ) = { ϕ ′ ( z 0 ) n : n = 0 , 1 , 2 , … } ∪ { 0 } (2)\sigma ({C_\phi }) = \{ \phi ’{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\} .