$V$ is Banach space, invertible operators $T_n \to -I$ in $B(V) \implies T^{-1}_n → -I$.

Question

Im stuck at this problem. I know that $f_n(f^{-1}_n+I)=(I+f_n)$. But I can't seem to use this fact. Any hints?

P.s. I tried to use some epsilon proof but I got stuck.

Answer 1

Note that $$\|T_n^{-1}+I\|\le \|T_n^{-1}\|\|T_n+I\|,$$ so it suffices to show $\|T_n^{-1}\|$ is bounded as $n\to\infty$. We find that for every $n$ with $\|T_n+I\|<1$, $$ T_n^{-1}=-\sum_{k=0}^\infty (T_n+I)^k $$ and this easily implies that $$ \|T_n^{-1}\|\le \sum_{k=0}^\infty \|T_n+I\|^k \le \sum_{k=0}^\infty 2^{-k}=2 $$ for every $n$ such that $\|T_n+I\|\le \frac12$.