The zero-product problem for Toeplitz operators with radial symbols

TL;DR

This paper proves that for Toeplitz operators with radial symbols on the Bergman space, the zero-product property holds, meaning if their product is zero, then at least one symbol must be zero, under certain conditions.

Contribution

The paper establishes the zero-product property for Toeplitz operators with radial symbols, extending understanding in cases where all but possibly one symbol are radial.

Findings

Zero-product property holds for Toeplitz operators with radial symbols.

The result applies when all but possibly one symbol are radial functions.

The general case remains unresolved.

Abstract

For any bounded measurable function $f$ on the unit ball $B_n$, let $T_f$ be the Toeplitz operator with symbol $f$ acting on the Bergman space $A^2(B_n)$. The Zero-Product Problem asks: if $f_1,..., f_N$ are bounded measurable functions such that $T_{f_1}... T_{f_N}=0$, does it follow that one of the functions must be zero almost everywhere? This paper give the affirmative answer to this question when all except possibly one of the symbols are radial functions. The answer in the general case remains unknown.