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## Invertible Operator [closed]

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? - 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