# 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)

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 $\Delta$.