Abstract We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on Lᵖ for every p>1, and of weak type (1, 1). We also prove necessary and sufficient conditions for the Lᵖ -boundedness of the extension of a class of Toeplitz-type operators.
Ottazzi et al. (Fri,) studied this question.