The commutant of a multiplication operator with a finite Blaschke product symbol on the Sobolev disk algebra

Abstract

Let R(D) be the algebra generated in the Sobolev space W22(D) by the rational functions with poles outside the unit disk D¯. This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator MB(z) on R(D) is studied, where B(z) is an n-Blaschke product. We prove that an operator A∈L(R(D)) is in A'(MB(z)) if and only if A=∑i=1nMhiMΔ(z)−1Ti, where {hi}i=1n⊂R(D), and Ti∈L(R(D)) is given by (Tig)(z)=∑j=1n(−1)i+jΔij(z)g(Gj−1(z)), i=1,2,…,n, G0(z)≡z.