We study the zero product problem of Toeplitz operators on the Hardy space and Bergman space over an annulus. Assuming a condition on the Fourier expansion of the symbols, we show that there are no zero divisors in the class of Toeplitz operators on the Hardy space of the annulus. Using the reduction theorem due to Abrahamse, we characterize compact Hankel operators on the Hardy space of the annulus, which also leads to a zero product result. Similar results are proved for the Bergman space over the annulus.

We are grateful to the referee for a careful reading of the manuscript and for making several suggestions which has improved the exposition of the paper substantially. The research of the first-named author is supported by NBHM Postdoctoral Fellowship at the Indian Institute of Science, Bangalore, India.