fa.functional analysis – Lipschitz estimate for compact operators

Let $T,S$ be two positive compact operators with zero null space.

When can I expect an estimate of the form

$$Vert T^{1/2}
S^{-1}T^{1/2}-1Vert le C Vert T-SVert_Y?$$

I intentionally write $Y$ on the right hand side to indicate that for such a bound to hold, one probably has to penalize small eigenvalues of $S$?

Is there perhaps a better way to quantify that if $T$ and $S$ are close, then $Vert T^{1/2}
is close to zero?