Resolution of the identity

A one-parameter family $ \{ E _ \lambda \} $, $ - \infty < \lambda < \infty $, of orthogonal projection operators acting on a Hilbert space $ {\mathcal H} $, such that

1) $ E _ \lambda \leq E _ \mu $ if $ \lambda < \mu $;

2) $ E _ \lambda $ is strongly left continuous, i.e. $ E _ {\lambda - 0 } = E _ \lambda $ for every $ \lambda \in ( - \infty , \infty ) $;

3) $ E _ \lambda \rightarrow 0 $ as $ \lambda \rightarrow - \infty $ and $ E _ \lambda \rightarrow E $ as $ \lambda \rightarrow \infty $; here 0 and $ E $ are the zero and the identity operator on the space $ {\mathcal H} $.

Condition 2) can be replaced by the condition of strong right continuity at every point $ \lambda \in ( - \infty , \infty ) $.

Every self-adjoint operator $ A $ acting on $ {\mathcal H} $ generates in a unique way a resolution of the identity. Here, in addition to 1)–3), the following conditions also hold:

4) if $ B $ is a bounded operator such that $ B A = A B $, then $ B E _ \lambda = E _ \lambda B $ for any $ \lambda $;

5) if $ A $ is a bounded operator and $ m $, $ M $ are its greatest lower and least upper bounds, respectively, then

$$ E _ \lambda = 0 \textrm{ for } - \infty < \lambda < m \ \ \textrm{ and } \ E _ \lambda = E \textrm{ for } M < \lambda < \infty . $$

The resolution of the identity given by the operator $ A $ completely determines the spectral properties of that operator, namely:

a) a point $ \lambda $ is a regular point of $ A $ if and only if it is a point of constancy, that is, if there is a $ \delta > 0 $ such that $ E _ \mu = E _ \lambda $ for $ \mu \in ( \lambda - \delta , \lambda + \delta ) $;

b) a point $ \lambda _ {0} $ is an eigenvalue of $ A $ if and only if at this point $ E _ \lambda $ has a jump, that is, $ E _ {\lambda _ {0} + 0 } - E _ {\lambda _ {0} } > 0 $;

g) if $ E ( \Delta ) = E _ \mu - E _ \lambda $, then $ L _ {E ( \Delta ) } = E ( \Delta ) {\mathcal H} $ is an invariant subspace of $ A $.

Hence the resolution of the identity determined by the operator $ A $ is also called the spectral function of this operator (cf. Spectral resolution).

Conversely, every resolution of the identity $ \{ E _ \lambda \} $ uniquely determines a self-adjoint operator $ A $ for which this resolution is the spectral function. The domain of definition $ D ( A) $ of $ A $ consists exactly of those $ x \in {\mathcal H} $ for which

$$ \int\limits _ {- \infty } ^ \infty \lambda ^ {2} d \langle E _ \lambda x , x \rangle < \infty , $$

and there is a representation of $ A $ as an operator Stieltjes integral:

$$ A = \int\limits _ {- \infty } ^ \infty \lambda d E _ \lambda . $$

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 $ \gamma $) mentioned above one may add that the spectrum of the restriction of $ A $ to $ L _ {E( \Delta ) } $ is contained in the set $ \Delta $.