Let $T$ be an unbounded self-adjoint operator on a Hilbert space and let $E(\lambda )$ be the associated spectral measure and $R(\lambda ) = (T-\lambda )^{-1}$ the resolvent. By Stone's theorem we have $$ (1) \qquad \frac{dE}{d\lambda } = \frac{1}{2\pi i} (R(\lambda + i0) - R(\lambda - i0)), $$ where $R(\lambda + i0) = \lim _{\varepsilon \searrow 0}R(\lambda + i\varepsilon )$ etc.

Assume I also know an operator $U_{\lambda }$ such that $$ U_{\lambda }TU_{\lambda }^{-1} = \lambda $$ is multiplication by $\lambda $.

Can I somehow combine these identities to express the right hand side of (1) in terms of $U_{\lambda }$?