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:

References

[1]	F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[2]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in a Hilbert space" , 1–2 , F. Ungar (1961–1963) (Translated from Russian)
[3]	L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)

Comments

To the property ) mentioned above one may add that the spectrum of the restriction of to is contained in the set .