The commutant and similarity invariant of analytic Toeplitz operators on Bergman space

Abstract

Famous von Neumann-Wold Theorem tell us that each analytic Toeplitz operator with $n+1$-Blaschke factors is unitary to $n+1$ copies of unilateral shift on Hardy space. It is obvious that von Neumann-Wold Theorem does not hold in Bergman space. In this paper, using the basis constructed by Michael Stessin and Kehe Zhu (see[1]) on Bergman space we prove that each analytic Toeplitz operator $M_{B(z)}$ is similar to $n+1$ copies of Bergman shift if and only if $B(z)$ is a $n+1$-Blaschke product. From above theorem, we c