Vol. 162, No. 2, 1994

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Commutants of Toeplitz operators on the Bergman space

Zeljko Cuckovic

Vol. 162 (1994), No. 2, 277–285

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.

Mathematical Subject Classification 2000
Primary: 47B35
Secondary: 47B38
Received: 17 April 1992
Revised: 23 October 1992
Published: 1 February 1994
Zeljko Cuckovic
Department of Mathematics
University of Toledo
Toledo OH 43606
United States