A note on composition operators on model spaces

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankhah we consider the following problem: for a given inner function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>∉</mml:mo> <mml:mi mathvariant="sans-serif">Aut</mml:mi> <mml:mo>(</mml:mo> <mml:mi>𝔻</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , find a non-constant inner function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> satisfying the functional equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>τ</mml:mi> <mml:mi>Ψ</mml:mi> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> is a unimodular constant. We prove that this problem has a solution if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> satisfying the equation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . If in addition, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> is parabolic, we prove that the problem has a solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> for any unimodular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> . Finally, we show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> is of zero hyperbolic step, then no non-constant Bloch function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> and no unimodular constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>τ</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> .