Let [itex]T[/itex] be a (possibly unbounded) self-adjoint operator on a Hilbert space [itex]\mathscr H[/itex] with domain [itex]D(T)[/itex], and let [itex]\lambda \in \rho(T)[/itex]. Then we know that [itex](T-\lambda I)^{-1}[/itex] exists as a bounded operator from [itex]\mathscr H[/itex] to [itex]D(T)[/itex]. Question: do we also know that [itex](T-\lambda I)^{-1}[/itex] is self-adjoint? Can someone prove or give a counterexample?(adsbygoogle = window.adsbygoogle || []).push({});

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# Are resolvents for self-adjoint operators themselves self-adjoint?

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