# 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 $$Vert T^{1/2} S^{-1}T^{1/2}-1Vert$$ is close to zero?