Brown-Halmos type theorems for Toeplitz operators on the Bergman space of the upper half-plane

We investigate Brown-Halmos type theorems which consist in characterizing when the product of two Bergman space Toeplitz operators is a Toeplitz operator whose symbol is the product of their symbols. The main involved technique is the characterization of functions that are in the range of the Berezin transform, and those which have finite rank Berezin symbols. We establish the upper half-plane analogues of the results of P. Ahern and & Zcaron;. & Ccaron;u & ccaron;kovi & cacute; (2001), P. Ahern (2004), and N.V. Rao (2018). In particular, we consider two classes of Toeplitz operators, namely those with bounded harmonic symbols and those with bi-analytic or tri-analytic symbols. Due to the unbounded character of the domain, many challenging difficulties occur, requiring more careful analysis. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.