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''of a linear (dissipative) operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211001.png" /> with domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211002.png" /> dense in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211003.png" />''
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$#C+1 = 43 : ~/encyclopedia/old_files/data/C021/C.0201100 Cayley transform
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The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211004.png" />, defined on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211005.png" />. A matrix version of this transform was considered by A. Cayley. The Cayley transform establishes a correspondence between the properties of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211006.png" /> with spectra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211007.png" />  "close"  to the real line and operators with almost-unitary spectra (close to the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211008.png" />). The following propositions are true: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c0211009.png" /> is a linear [[Dissipative operator|dissipative operator]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110010.png" /> is a contraction (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110012.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110013.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110014.png" /> is a contraction, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110016.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110018.png" /> for some linear dissipative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110019.png" />; in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110020.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110021.png" /> is symmetric (self-adjoint) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110022.png" /> is isometric (unitary); 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110024.png" />, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110025.png" /> is bounded if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110026.png" />; and 5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110027.png" /> is an operator ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110029.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110030.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110031.png" /> are bounded operators, then the converse is also valid: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110032.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110033.png" />. The Cayley transform also establishes a correspondence between certain other characteristics of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110035.png" />: classifications of parts of the spectrum, multiplicities of spectra, structures of invariant subspaces, functional calculi, spectral decompositions, etc. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110036.png" /> is a [[Self-adjoint operator|self-adjoint operator]] with resolution of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110040.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110041.png" />, is a resolution of the identity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110042.png" /> and
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021100/c02110043.png" /></td> </tr></table>
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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 \} $
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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 $;  
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3)  $  A $
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is symmetric (self-adjoint) if and only if  $  C _ {A} $
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is isometric (unitary); 4)  $  \sigma ( A) = \omega ( \sigma ( C _ {A} )) $,
 +
where  $  \omega ( \zeta ) = i ( 1 + \zeta ) ( 1 - \zeta )  ^ {-} 1 $,
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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====
 
====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>
 
<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>

Revision as of 16:43, 4 June 2020


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, 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 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

[1] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[2] B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)
How to Cite This Entry:
Cayley transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=46290
This article was adapted from an original article by N.K. Nikol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article