Journal of the
Korean Mathematical Society JKMS

Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces and let $T, \widetilde{T}=T+\delta T$ be bounded operators from $\mathcal{H}$ into $\mathcal{K}$. In this article, two facts related to the perturbation bounds are studied. The first one is to find the upper bound of $\|\widetilde{T}^+-T^+\|$, which extends the results obtained by the second author and enriches the perturbation theory for the Moore-Penrose inverse. The other one is to develop explicit representations of projectors $\|\widetilde{T}\widetilde{T}^+-TT^+\|$ and $\|\widetilde{T}^+\widetilde{T}-T^+T\|$. In addition, some spectral cases related to these results are analyzed.

Keywords: generalized inverse, Moore-Penrose inverse, perturbation, block operator matrix

MSC numbers: 47A05, 46C07, 15A09