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hi,

Let $C,T$ be bounded self-adjoint operators on a Hilbert space $\mathcal{H}$ with the property that $0 \leq T - C$ and the spectrum of $T$ and $C$ is positive real (witout containing zero). Do we have that $0 \leq T(1 + \lambda \cdot T)^{-1} - C(1 + \lambda \cdot C)^{-1}$, where $\lambda > 0$ ? if Yes, how is this to prove and if no can one give me a counterexample please. Tanks in advance.

denis

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It might help if you write $T(1 + \lambda T)^{-1} = \lambda^{-1} - \lambda^{-1}(1 + \lambda T)^{-1}$. – Jesse Peterson Dec 28 2011 at 11:58
Why do you ask? (by which I mean: is this homework?) Also: are you also peter? – Igor Rivin Dec 28 2011 at 11:58
Seconding the first part of Igor's comment: is this from an exercise, or a step in a paper that you are trying to understand? In either case, a reference link or description might help to provide extra context – Yemon Choi Dec 28 2011 at 13:45