I am interested to find an approximate formula for $$A (A+B)^{-1} A\ ,$$ for two positive matrices $A$ and $B$ whose supports are almost orthogonal. If the support of $A$ and $B$ are orthogonal then, $$A (A+B)^{-1} A=A .$$
The question is how this formula is modified if the support of $A$ and $B$ are almost orthogonal. More specifically, let $A$ and $B$ be two positive matrices with trace one, i.e. two density operators. Suppose the $l_1$-norm of $A-B$ is larger than or equal to $2-\epsilon$, i.e. $$\|A-B\|_1\ge 2-\epsilon\ $$ where $\epsilon \ge 0.$ Then, for $\epsilon=0$, the support of $A$ and $B$ are orthogonal, which implies$A (A+B)^{-1} A=A .$
The $\textbf{question}$ is how this formula is corrected for nonzero, but small $\epsilon>0$. Can we find a bound on $\|A (A+B)^{-1} A-A \|$?
