Zero Products of Toeplitz operators on the Hardy and Bergman spaces over an annulus

TL;DR

This paper investigates zero-product (zero-divisor) phenomena for Toeplitz operators on Hardy and Bergman spaces over the annulus $A_{1,R}$. It leverages Abrahamse’s reduction theorem to relate annulus problems to the unit disk and employs Mellin-transform analysis of Fourier components to derive zero-product results; it also provides a Hartman-type characterization of compact Hankel operators on the annulus and uses it to deduce corresponding zero-product consequences. The results establish no-zero-divisors results for the Hardy annulus under suitable Fourier-Mellin hypotheses and prove analogous statements for the Bergman annulus with explicit quasi-homogeneous symbols, highlighting the role of the reduction technique in multi-connected settings.

Abstract

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.

Theorems & Definitions (7)

• proof
• proof
• proof
• proof
• proof
• proof
• proof