We introduce a sequence of positive linear operators involving adjoint Bernoulli polynomials of the first kind, and we focus on the approximation properties of these operators. One of the main objectives is to get estimates for the order of approximation by means of first-order modulus of continuity, the Lipschitz condition, first modulus of derivative and a combination of first-order modulus of continuity and extended second-order modulus. Further, we give Voronovskaya type and Grüss--Voronovskaya type asymptotic results. Finally, we give two examples for error estimation by using Maple software.