Perturbation analysis of $A_{T,S}^{(2)}$ on Hilbert spaces

In this paper, we investigate the perturbation analysis of $A_{T,S}^{(2)}$ when $T,\,S$ and $A$ have some small perturbations on Hilbert spaces. We present the conditions that make the perturbation of $A_{T,S}^{(2)}$ is stable. The explicit representation for the perturbation of $A_{T,S}^{(2)}$ and the perturbation bounds are also obtained.