The authors introduce a double sequence \((L_n^{\langle k \rangle}: n\geq 1,k\geq 0)\) of linear polynomial operators which includes, as particular cases, the Bernstein, Kantorovič and Cao operators. For the operators \(L_n^{\langle k\rangle}\) the authors discuss several approximation properties: the convergence properties, the preservation of global smoothness and classes of functions determined by concave moduli of continuity. A remarcable feature of our approach is that if \(f\) is differentiable, the approximation properties of both \(L_n^{\langle k\rangle}f\) and its derivatives can be discussed simultaneously.