This paper describes the
commutants of certain analytic Toeplitz operators. To underline the difference
between the Bergman and Hardy spaces, we first prove that on the Bergman space
La2 the only isometric Toeplitz operators with harmonic symbols are scalar multiples
of the identity. If T denotes the norm closed subalgebra of L(La2) generated by
Toeplitz operators, we show that for each positive integer n, {Tzn}′∩ T is the set of
all analytic Toeplitz operators. This result is also valid for the Hardy space.
Here {Tzn}′ denotes the commutant of Tzn. Finally we prove the analogous
result for Tun, where u is an analytic, one-to-one map of the unit disk onto
itself.
