Algebraic and spectral properties of H-Toeplitz operators on the Bergman space

In this paper, we discuss the zero product problem of two H-Toeplitz operators and the commuting problem of H-Toeplitz and Hankel operators on the Bergman space. We establish necessary and sufficient conditions for the product of an H-Toeplitz operator and a Hankel operator equals another H-Toeplitz (Hankel) operator for a certain class of symbols. Moreover, we study the spectrum of the H-Toeplitz operators $ B_{z^{N}} $ and $ B_{\overline{z}^{N}} $, where $ N $ is a non-negative integer.