# Spectral measure

A unitary homomorphism from some Boolean algebra of sets into the Boolean algebra of projection operators on a Banach space. Every operator $T$ on a Banach space $X$ defines a spectral measure on the set of open-and-closed subsets of its spectrum $\sigma(T)$ by the formula $$E(\alpha) = \frac{1}{2 \pi i} \int_\Gamma (zI-T)^{-1} dz \ ,$$ where $\Gamma$ is a Jordan curve separating $\alpha$ from $\sigma(T) \setminus \alpha$. Here, $TE(\alpha) = E(\alpha)T$ and $\sigma\left(T \downharpoonright_{E(\alpha)X}\right) \subseteq \bar\alpha$. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the spectral theory of linear operators.