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In this paper we firstly introduce the concepts of capacity and Cegrell'sclasses associated to any $m$-positive closed current $T$. Next, afterinvestigating the most imporant related properties, we study the definition andthe continuity of the complex hessian operator in several cases, generalizingthen the work of Demailly and Xing in this direction. We also prove a Xing-typecomparison principle for the analogous Cegrell class $\mathcal{F}^{m,T}$ ofnegative $m$-subharmonic functions. Finally, we generalize the work of BenMessaoud-El Mir on the complex Monge-Amp\`ere operator and the Lelong-Skodapotential associated to a positive closed current.

Complex Hessian Operator, $m$-capacity, Cegrell's classes and $m$-Potential associated to a Positive Closed Current