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In this paper, we first study the definition and the continuity of thecomplex Hessian operator associated to an $m$-positive closed current $T$, forsome classes of unbounded $m$-subharmonic functions as well as when we considera regularization sequence of $T$. Next, we introduce the notion of weighted$(m,T)$-capacity in the complex Hessian setting and we investigate the linkwith the weighted $m$-extremal function. As an application we give acharacterization of the Cegrell classes ${\mathscr F}^m$ and ${\mathscr E}^m$by means of the weighted $(m,1)$-capacity. Furthermore, we prove a subsolutiontheorem for a general complex Hessian equation relatively to $T$.

Complex Hessian Operator associated to an $m$-positive closed current and weighted $m$-capacity