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If M is a bounded operator acting on a Hilbert space, with the elements of the spectrum of the operator M + M^* (where M^$ is the adjoint of M) strictly positive, then is the operator M + M^* invertible?

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 By definition, if $0$ is not in the spectrum of the bounded operator $A$, then $A$ is invertible. – Johannes Ebert Mar 10 2011 at 19:18
Isn't the definition of the spectrum? Take M self-adjoint then what you say implies the spectrum is a subset of the interval $(0,\infty)$ and therefore does not include zero. By definition this means zero is in the resolvant set and $M=M-0Id$ is invertible with bounded inverse. – Abdelmalek Abdesselam Mar 10 2011 at 19:22
Is this an exercise? Or homework? – András Bátkai Mar 10 2011 at 20:57

closed as too localized by Johannes Ebert, Yemon Choi, Andres Caicedo, Andreas Thom, Bill Johnson Mar 10 2011 at 22:38

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