Perturbation analysis for Moore-Penrose inverse of closed operators on Hilbert spaces

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 ${\bar{T}}^{\mathrm{†}}$ and the upper bounds of $\mathrm{\parallel }{\bar{T}}^{\mathrm{†}}\mathrm{\parallel }$ and $\mathrm{\parallel }{\bar{T}}^{\mathrm{†}}-T^{\mathrm{†}}\mathrm{\parallel }$. These results obtained in this paper extend and improve many related results in this area.