  Let $E$ be a Hermitian vector bundle over a complete K\"{a}hler manifold
$(X,\omega)$, $\dim_{\mathbb{C}}X=n$, with a $d$(bounded) K\"{a}hler form
$\omega$, $d_{A}$ be a Hermitian connection on $E$. The goal of this article is
to study the $L^{2}$-Hodge theory on the vector bundle $E$. We extend the
results of Gromov's \cite{Gro} to the Hermitian vector bundle. At last, as an
application, we prove a gap result for Yang-Mills connection on the bundle $E$
over $X$.
