# Spectral resolution

A monotone mapping $P(\cdot)$ from the real line into the set of orthogonal projectors on a Hilbert space, which is left-continuous in the strong operator topology and satisfies the conditions $$\lim_{t \rightarrow -\infty} P(t) = 0 \ ;\ \ \ \lim_{t \rightarrow +\infty} P(t) = I \ .$$ Every self-adjoint (i.e. taking self-adjoint values) strongly countably-additive Borel spectral measure $E(\cdot)$ on the line defines a spectral resolution by the formula $P(t) = E((-\infty,t))$, and for every spectral resolution there is a unique spectral measure defining it.
The concept of a spectral resolution is fundamental in the spectral theory of self-adjoint operators: By the spectral decomposition theorem (cf. Spectral decomposition of a linear operator), every such operator has an integral representation $\int_{-\infty}^{\infty} t dP(t)$, where $P(t)$ is some spectral resolution. An analogous role in the theory of symmetric operators is played by the concept of a generalized spectral resolution, which is a mapping from the real line into the set of non-negative operators that satisfies all the conditions imposed on spectral resolutions, except that the values need not be projectors. Every generalized spectral resolution can be extended to a spectral resolution on a larger space (Naimark's theorem).