  In this paper we firstly introduce the concepts of capacity and Cegrell's
classes associated to any $m$-positive closed current $T$. Next, after
investigating the most imporant related properties, we study the definition and
the continuity of the complex hessian operator in several cases, generalizing
then the work of Demailly and Xing in this direction. We also prove a Xing-type
comparison principle for the analogous Cegrell class $\mathcal{F}^{m,T}$ of
negative $m$-subharmonic functions. Finally, we generalize the work of Ben
Messaoud-El Mir on the complex Monge-Amp\`ere operator and the Lelong-Skoda
potential associated to a positive closed current.
