  In this paper, we investigate the perturbation for the Moore-Penrose inverse
of closed operators on Hilbert spaces. By virtue of a new inner product defined
on $H$, we give the expression of the Moore-Penrose inverse $\bar{T}^\dag$ and
the upper bounds of
  $\|\bar{T}^\dag\|$ and $\|\bar{T}^\dag -T^\dag\|$. These results obtained in
this paper extend and improve many related results in this area.
