In this article, we establish the inclusion relationship between the kernel and range of little Hankel operators and Toeplitz operators on the Bergman space. Further, we expound on the conditions for finite rank operators and invertibility of operators on \(A^2(\mathbb {D}).\) Moreover, we explore some necessary and sufficient conditions under which the bounded linear operators on the Bergman space \( A^2(\mathbb {D})\) are positive.