View source for Resolution of the identity

<!--
r0815901.png
$#A+1 = 51 n = 0
$#C+1 = 51 : ~/encyclopedia/old_files/data/R081/R.0801590 Resolution of the identity
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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

====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 $.