Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space. We then study $$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$

I would assume that this norm is bounded by one.

But I fail to see how one could actually show this? Cause the definition of the square root using the functional calculus is rather abstract.