A Banach algebra structure on the $q$-Bergman space and related topics

We show that the $q$-Bergman space with the Duhamel product has a Banach algebra structure and describe its nontrivial closed ideals. Moreover, we discuss Katznelson–Tzafriri type theorems in the $q$-Bergman space. We also give applications to the Volterra integral operator on this space and to some integral equations.