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

Abstract

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofMZ2 (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantMZ2 ifSMZ2k+1−MZ2k+1S is compact for some nonnegative integerk, thenS=Mϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofMZn defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formMϕ.