View source for Spectral resolution

''spectral function, resolution of the identity''

A monotone mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864701.png" /> which is left-continuous in the strong operator topology, from the real line into the set of orthogonal projectors on a Hilbert space, and satisfying the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864702.png" /></td> </tr></table>

Every self-adjoint (i.e. taking self-adjoint values) strongly countably-additive Borel [[Spectral measure|spectral measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864703.png" /> on the line defines a spectral resolution by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864704.png" />, 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|Spectral decomposition of a linear operator]]), every such operator has an integral representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864706.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>