For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,q+1}(V)$ which are nilpotent just like the Dolbeault operator. My question is if there is a natural way to generalize the Lefschetz/dual Lefschetz operator to bundle-valued forms. I'm particularly interested if there is an expression for the generalized dual Lefschetz operator on $(0,1)$-$V$ valued forms.
I guess one could just define $L_V = L \otimes I$ and $\Lambda_V = \Lambda \otimes I$ on $\Omega^{p,q}(V)$, but was wondering if there is anything else in the literature.