I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible operators on a Hilbert space $\mathcal{H}$ such that $C \leq T$. Then it follows also that $T^{-1} \leq C^{-1}$. Or maybe this is even not true? If this is not true can somebody give me a counter-example or if it is true some strategy how to solve this? I would be very thankful.
mika