Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let $S$ be any other operator in $B(H)$. Then I want to show that if $\epsilon\rightarrow 0$, then $$\|T_\epsilon S T_\epsilon -T^{-\frac{1}{2}}ST^{-\frac{1}{2}}\|\rightarrow 0.$$
I am not sure if $T^{-\frac{1}{2}}$ is always defined. What should be the starting point? Hints are most welcome.