Douglas question for general harmonic symbols on the Bergman space

Determine whether the Toeplitz operator T_\varphi on the Bergman space L_a^2 is invertible for every bounded harmonic function \varphi on the unit disk satisfying |\varphi(z)|\geq\delta>0 for all z\in\mathbb D.

Background

The paper discusses the Bergman-space analogue of the Douglas question: whether a Toeplitz operator T_\varphi is invertible whenever the Berezin transform of its symbol is bounded away from zero. For bounded harmonic symbols, the Berezin transform coincides with the symbol itself, so the condition becomes |\varphi(z)|\geq\delta>0 throughout the unit disk.

Several affirmative results are known for special classes of harmonic symbols, including real harmonic functions, certain harmonic polynomials, symbols of the form \overline{z}+p, and symbols involving bounded analytic functions. The paper proves another affirmative result for symbols of the form \overline{z}+h with h in the disk algebra. It nevertheless identifies the question for general harmonic symbols as unresolved.

References

Although the Douglas question on the Bergman space $L_a2$ remains open for harmonic symbols, significant progress has been achieved in the study of the invertibility of Toeplitz operators with various symbols on $L_a2$.

— The Douglas question for functions of the form $\overline{z}+h$ with $h$ in the disk algebra  (2608.22811 - Wang et al., 24 Aug 2026) in Section 1, Introduction