Spectral resolution

spectral function, resolution of the identity

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).

References

[1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) Zbl 0467.47001
[2]	M.A. Naimark, "Self-adjoint extensions of the second kind of a symmetric operator" Izv. Akad. Nauk SSSR Ser. Mat. , 4 : 1 (1940) pp. 53–104 (In Russian) (English abstract) Zbl 0025.06402