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''of a linear (dissipative) operator  $  A $
with domain  $   \mathop{\rm Dom}  A $
dense in a Hilbert space  $  H $''

The operator  $  C _ {A} = ( A - iI) ( A + iI)  ^ {-1} $, 
defined on the subspace  $   \mathop{\rm Dom}  C _ {A} = ( A + iI)  \mathop{\rm Dom}  A $. 
A matrix version of this transform was considered by A. Cayley. The Cayley transform establishes a correspondence between the properties of operators  $  A $
with spectra  $  \sigma ( A) $"
close"  to the real line and operators with almost-unitary spectra (close to the circle  $  \{ {\zeta \in \mathbf C } : {| \zeta | = 1 } \} $). 
The following propositions are true: 1) if  $  A $
is a linear [[Dissipative operator|dissipative operator]], then  $  C _ {A} $
is a contraction (i.e.  $  \| C _ {A} x \| \leq  \| x \| $, 
$  x \in  \mathop{\rm Dom}  A $) 
and  $   \mathop{\rm Ker} ( I - C _ {A} ) = \{ 0 \} $; 
2) if  $  T $
is a contraction,  $   \mathop{\rm Ker} ( I - T) = \{ 0 \} $
and  $  ( I - T)  \mathop{\rm Dom}  T $
is dense in  $  H $, 
then  $  T = C _ {A} $
for some linear dissipative operator  $  A $; 
in fact,  $  A = i ( I + T) ( I - T)  ^ {-1} $; 
3)  $  A $
is symmetric (self-adjoint) if and only if  $  C _ {A} $
is isometric (unitary); 4)  $  \sigma ( A) = \omega ( \sigma ( C _ {A} )) $, 
where  $  \omega ( \zeta ) = i ( 1 + \zeta ) ( 1 - \zeta )  ^ {-1} $, 
in particular,  $  A $
is bounded if and only if  $  1 \notin \sigma ( C _ {A} ) $; 
and 5) if  $  \gamma $
is an operator ideal in  $  H $, 
then  $  A - B \in \gamma $
implies  $  C _ {A} - C _ {B} \in \gamma $; 
if  $  A, B $
are bounded operators, then the converse is also valid:  $  C _ {A} - C _ {B} \in \gamma $
implies  $  A - B \in \gamma $. 
The Cayley transform also establishes a correspondence between certain other characteristics of the operators  $  A $
and  $  C _ {A} $: 
classifications of parts of the spectrum, multiplicities of spectra, structures of invariant subspaces, functional calculi, spectral decompositions, etc. Thus, if  $  A $
is a [[Self-adjoint operator|self-adjoint operator]] with resolution of the identity  $  \{ E _ {t} \} $, 
$  t \in \mathbf R $, 
then  $  \{ F _ {s} \} $, 
$  F _ {s} = E _ {t} $
for  $  s = - 2   \mathop{\rm arctan}  t $, 
is a resolution of the identity for  $  C _ {A} $
and

$$ 
A  = \ 
\int\limits _ { 0 } ^ { {2 }  \pi }
 \mathop{\rm cotan} \ 
{
\frac{s}{2}
 }  dF _ {s} .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Szökefalvi-Nagy,   Ch. Foiaş,   "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR></table>