Spectral integral: verification of my conceptual understanding
Solution 1:
Yes, that's the standard way to do it. The only thing I would mention (or ask you to clarify, if I were evaluating you) is when you write $$ t\,(1-E(\epsilon))=\int_{\sigma(t)}\chi^{\phantom{A}}_{(\epsilon,\infty)}\,\lambda\,dE(\lambda) $$ you are using the extremely important fact that the map $$f\longmapsto\int_{\sigma(t)}f(\lambda)\,dE(\lambda)$$ is a $*$-homomorphism. One is used to integrals being linear, but being multiplicative is a very particular property that depends on the fact that $E$ is projection-valued.