Roots of Toeplitz Operators on the Bergman Space

One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex plane C is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In [4], the first author obtained a complete description of the commutant of Toeplitz operator T with any quasihomogeneous symbol φ(r)e , p > 0 in case it has a Toeplitz p-th root S with symbol ψ(r)e , namely, commutant of T is the closure of the linear space generated by powers S which are Toeplitz. But the existence of p-th root was known until now only when φ(r) = r, m ≥ 0. In this paper we will show the existence of p-th roots for a much larger class of symbols, for example, it includes such symbols for which