Math 6410 • ODE

Let \(A\) be a selfadjoint operator in a Hilbert space \(\mathcal {H}\) and let \(\{E_\lambda \}\) be the spectral family of \(A\). Prove that if a real number \(\mu \) is in the resolvent set of \(A\) then there exists an interval \(I=\{\lambda :\lvert \lambda -\mu \rvert <\delta \}\) such that \(E_\lambda =\text {const.}\) for all \(\lambda \in I\). (You may use without proof the Stieltjes-Hellinger-Stone theorem: \[ R_z=(A-z)^{-1}. \] \[ \bigl ((E_{b+}+E_{b-}-E_{a+}-E_{a-})u,u\bigr ) =\lim _{\epsilon \to 0+}\frac {1}{\pi i} \int _a^b\bigl ((R_{\lambda +i\epsilon }-R_{\lambda -i\epsilon })u,u\bigr )\,d\lambda . \]