Resolution of the identity
A one-parameter family , , of orthogonal projection operators acting on a Hilbert space , such that
1) if ;
2) is strongly left continuous, i.e. for every ;
3) as and as ; here 0 and are the zero and the identity operator on the space .
Condition 2) can be replaced by the condition of strong right continuity at every point .
Every self-adjoint operator acting on generates in a unique way a resolution of the identity. Here, in addition to 1)–3), the following conditions also hold:
4) if is a bounded operator such that , then for any ;
5) if is a bounded operator and , are its greatest lower and least upper bounds, respectively, then
The resolution of the identity given by the operator completely determines the spectral properties of that operator, namely:
a) a point is a regular point of if and only if it is a point of constancy, that is, if there is a such that for ;
b) a point is an eigenvalue of if and only if at this point has a jump, that is, ;
g) if , then is an invariant subspace of .
Hence the resolution of the identity determined by the operator is also called the spectral function of this operator (cf. Spectral resolution).
Conversely, every resolution of the identity uniquely determines a self-adjoint operator for which this resolution is the spectral function. The domain of definition of consists exactly of those for which
and there is a representation of as an operator Stieltjes integral:
|||F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)|
|||N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in a Hilbert space" , 1–2 , F. Ungar (1961–1963) (Translated from Russian)|
|||L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)|
To the property ) mentioned above one may add that the spectrum of the restriction of to is contained in the set .
Resolution of the identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolution_of_the_identity&oldid=13743