Abstract. The commutants of analytic Toeplitz operators have been studied in depth. One of the main results in this area is that certain Toeplitz operators have commutants generated by the Toeplitz operators and the composition operators that commute with it. This result gives rise to a question about the commutant of a composition operator: Is the commutant of a composition operator generated by those composition operators and multiplication operators that commute with it? Certain examples suggest a modified question: Is the commutant of a composition operator generated by the multiplication operators that commute with it and the composition operator itself? We will give examples where this is not the case by constructing composition operators with commutants larger than this.

AMS Subject Classification (1991): 47B38

Received November 28, 2000, and in revised form July 3, 2001. (Registered under 2849/2009.)