A central issue of the present paper is to consider the Korovkin-type approximation properties of the Lupaş-Jain operators using $A$-statistical convergence and Abel convergence, which are well-known summability methods. We provide an instance of a sequence of positive linear operators to which the weighted Korovkin-type theorem does not apply, but the $A$-statistical approximation theorem does. Our results are one-way more substantial than some approximation results given in [Başcanbaz-Tunca et al., <i>On Lupaş-Jain Operators</i>, Stud. Univ. Babeş-Bolyai Math., <b>63(4)</b> (2018), 525--537].