On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.