Commuting Toeplitz operators with harmonic symbols on the Fock space

We give a counterexample to a characterization of commuting Toeplitz operators due to Bauer, Choe, and Koo [JFA, 268 (2015), 3017–3060]. We prove a corrected characterization for Toeplitz operators with harmonic symbols of exponential type on the Fock space in one complex variable.