Many-objective optimization (four or more objectives) presents many challenges to be considered, highlighting the need to create better algorithms prepared to deal efficiently with the increasing number of objectives. One such challenge is to determine the most efficient operator or combination of operators to be used during the optimization. In order to deal with this challenge, we propose the use of adaptive operator selection mechanisms in many-objective optimization algorithms. Two adaptive operator selection mechanisms, Adaptive Pursuit (AP) and Probability Matching (PM), are incorporated into the NSGA-III framework (a recently proposed, state-of-the-art algorithm to solve many-objective problems) to autonomously select the most suitable operator while solving a many-objective problem, according to the previous performance of each operator. The proposed algorithms, NSGA-IIIAP and NSGA-IIIPM, are tested in four different multi-objective problems from the DTLZ test suite with 3 up to 20 objectives. Statistical tests were performed to infer the significance of the results. The hypothesis that adaptive ways to select the operator to be applied during each stage of the evolutionary process is an effective way to improve the performance of the NSGA-III framework is corroborated by our results.