Lower bound for $\| (I+A)^{-1} – (I+B)^{-1}\|$ where $A, B \ge 0$

Question

Given two positive semi-definite matrices $A,B \ge 0$, I am interested in finding a lower bound for the operator norm:

$$
\| (I+A)^{-1} – (I+B)^{-1}\|.
$$

So far I have only been able to find an upper bound, namely

$$
\| (I+A)^{-1} – (I+B)^{-1}\| \le \| B-A\|
$$
which follows by noting that
$$
(I+A)^{-1} – (I+B)^{-1} = (I+A)^{-1}(B-A)(I+B)^{-1}
$$
and that $\| (I+A)^{-1} \| \le 1$ since $A \ge 0$ and similarly for the term involving $B$. I am stuck on finding a good lower bound though.

Answer 1

Your trick already gives you a lower bound. Since the operator norm is submultiplicative, \begin{aligned} &\quad\,\|I+A\|\|(I+A)^{-1}(B-A)(I+B)^{-1}\|\|I+B\|\\ &\ge\|(I+A)(I+A)^{-1}(B-A)(I+B)^{-1}(I+B)\|\\ &=\|B-A\|. \end{aligned} Therefore \begin{aligned} \|(I+A)^{-1}(B-A)(I+B)^{-1}\| \ge\frac{\|B-A\|}{\|I+A\|\|I+B\|} =\frac{\|B-A\|}{(1+\|A\|)(1+\|B\|)}. \end{aligned}