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A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334301.png" /> defined on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334302.png" /> which is dense in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334303.png" /> such that
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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/d/d033/d033430/d0334304.png" /></td> </tr></table>
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This requirement is sometimes replaced by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334305.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334306.png" />, i.e. the dissipativeness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334307.png" /> in this sense is equivalent to that of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334308.png" />.
+
A linear operator  $  A $
 +
defined on a domain  $  D _ {A} $
 +
which is dense in a Hilbert space  $  H $
 +
such that
  
A dissipative operator is said to be maximal if it has no proper dissipative extensions. A dissipative operator always has a closure, which also is a dissipative operator; in particular, a maximal dissipative operator is a closed operator. Any dissipative operator can be extended to a maximal dissipative operator. For a dissipative operator all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d0334309.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343010.png" /> belong to the resolvent set, and moreover
+
$$
 +
\mathop{\rm Im}  ( A x , x )  \geq  0 \  \textrm{ if }  x \in D _ {A} .
 +
$$
  
<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/d/d033/d033430/d03343011.png" /></td> </tr></table>
+
This requirement is sometimes replaced by the condition  $  \mathop{\rm Re}  ( A x , x ) \leq  0 $
 +
if  $  x \in D _ {A} $,
 +
i.e. the dissipativeness of  $  A $
 +
in this sense is equivalent to that of the operator  $  ( - iA ) $.
  
A dissipative operator is maximal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343014.png" />. An equivalent condition for maximality of a dissipative operator is that it is closed and that
+
A dissipative operator is said to be maximal if it has no proper dissipative extensions. A dissipative operator always has a closure, which also is a dissipative operator; in particular, a maximal dissipative operator is a closed operator. Any dissipative operator can be extended to a maximal dissipative operator. For a dissipative operator all points  $  \lambda $
 +
with $  \mathop{\rm Im}  \lambda < 0 $
 +
belong to the resolvent set, and moreover
  
<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/d/d033/d033430/d03343015.png" /></td> </tr></table>
+
$$
 +
\| A x - \lambda x \|  \geq  |  \mathop{\rm Im}  \lambda |  \| x \| ,
 +
\  x \in D _ {A} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343016.png" /> is a maximal [[Symmetric operator|symmetric operator]], then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343018.png" /> is a maximal dissipative operator. Dissipative and, in particular, maximal dissipative extensions may be considered for an arbitrary symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343019.png" />; their description is equivalent to the description of all maximal dissipative extensions of the conservative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343022.png" />.
+
A dissipative operator is maximal if and only if  $  ( A - \lambda I ) D _ {A} = H $
 +
for all  $  \lambda $
 +
with  $  \mathop{\rm Im}  \lambda < 0 $.  
 +
An equivalent condition for maximality of a dissipative operator is that it is closed and that
  
Dissipative operators are closely connected with contractions (cf. [[Contraction(2)|Contraction]]) and with the so-called accretive operators, i.e. operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343023.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343024.png" /> is a dissipative operator. In particular, an accretive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343025.png" /> is maximal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343026.png" /> is the generating operator (or generator) of a continuous one-parameter contraction semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343028.png" />. The Cayley transform
+
$$
 +
\mathop{\rm Im}  ( A  ^ {*} y , y ) \leq  0 ,\  y \in D _ {A  ^ {*}  } .
 +
$$
  
<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/d/d033/d033430/d03343029.png" /></td> </tr></table>
+
If  $  A _ {0} $
 +
is a maximal [[Symmetric operator|symmetric operator]], then either  $  A _ {0} $
 +
or  $  ( - A _ {0} ) $
 +
is a maximal dissipative operator. Dissipative and, in particular, maximal dissipative extensions may be considered for an arbitrary symmetric operator  $  A _ {0} $;  
 +
their description is equivalent to the description of all maximal dissipative extensions of the conservative operator  $  B _ {0} = iA _ {0} $:  
 +
$  \mathop{\rm Re}  ( B _ {0} x , x ) = 0 $,
 +
$  x \in D _ {B} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343030.png" /> is a maximal accretive operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343031.png" /> is a contraction not having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343032.png" /> as an eigen value, is used to construct the functional calculus and, in particular, the theory of fractional powers of maximal dissipative operators.
+
Dissipative operators are closely connected with contractions (cf. [[Contraction(2)|Contraction]]) and with the so-called accretive operators, i.e. operators  $  A $
 +
for which  $  iA $
 +
is a dissipative operator. In particular, an accretive operator $  A $
 +
is maximal if and only if  $  ( - A ) $
 +
is the generating operator (or generator) of a continuous one-parameter contraction semi-group  $  \{ T _ {s} \} _ {s \geq  0 }  $
 +
on  $  H $.  
 +
The Cayley transform
  
In the case of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343033.png" /> the definition of a dissipative operator is equivalent to the requirement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343035.png" /> is the imaginary part of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343036.png" />. For a completely-continuous dissipative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343037.png" /> on a separable Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343038.png" /> with nuclear imaginary part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343039.png" />, several criteria (i.e. necessary and sufficient conditions) for the completeness of the system of its root vectors are available; for example,
+
$$
 +
= ( A - I ) ( A + I )  ^ {-} 1 ,\ \
 +
= ( I + T ) ( I - T ) ^ {-} 1 ,
 +
$$
  
<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/d/d033/d033430/d03343040.png" /></td> </tr></table>
+
where  $  A $
 +
is a maximal accretive operator and  $  T $
 +
is a contraction not having  $  \lambda = 1 $
 +
as an eigen value, is used to construct the functional calculus and, in particular, the theory of fractional powers of maximal dissipative operators.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343041.png" /> are all eigen values of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343044.png" /> is the trace of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343045.png" /> (Livshits' criterion);
+
In the case of bounded linear operators  $  A $
 +
the definition of a dissipative operator is equivalent to the requirement  $  A _ {J} \geq  0 $,  
 +
where  $  A _ {J} = ( A  ^ {*} - A ) / 2 i $
 +
is the imaginary part of the operator $  A $.  
 +
For a completely-continuous dissipative operator  $  A $
 +
on a separable Hilbert space  $  H $
 +
with nuclear imaginary part  $  A _ {J} $,
 +
several criteria (i.e. necessary and sufficient conditions) for the completeness of the system of its root vectors are available; for example,
  
<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/d/d033/d033430/d03343046.png" /></td> </tr></table>
+
$$
 +
\sum _ {j = 1 } ^ {  \nu  ( A) }  \mathop{\rm Im}  \lambda _ {j} ( A)  = \
 +
\mathop{\rm tr}  A _ {J} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343047.png" /> is the real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343049.png" /> is the number of characteristic numbers of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343050.png" /> in the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343052.png" /> (Krein's criterion). The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343053.png" /> of eigen vectors corresponding to different eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343055.png" /> of a dissipative operator forms a basis of its closed linear span and is equivalent to an orthonormal basis if
+
where $  \lambda _ {j} ( A) $
 +
are all eigen values of the operator  $  A $,
 +
$  j = 1 \dots \nu ( A) \leq  \infty $,  
 +
and $  \mathop{\rm tr}  A _ {J} $
 +
is the trace of the operator $  A _ {J} $(
 +
Livshits' criterion);
  
<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/d/d033/d033430/d03343056.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\rho \rightarrow \infty } 
 +
\frac{n _ {+} ( \rho , A _ {R} ) } \rho
 +
  = 0 \  \textrm{ or } \  \lim\limits _ {\rho \rightarrow \infty }
 +
 +
\frac{n _ {-} ( \rho , A _ {R} ) } \rho
 +
  = 0 ,
 +
$$
  
The concept of a dissipative operator was also introduced for non-linear and even for multi-valued operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033430/d03343057.png" />. Such an operator on a Hilbert space is called dissipative if for any two of its values the inequality
+
where  $  A _ {R} = ( A + A  ^ {*} ) / 2 $
 +
is the real part of  $  A $,
 +
and  $  n _  \pm  $
 +
is the number of characteristic numbers of the operator $  A _ {R} $
 +
in the segment  $  [ 0 , \rho ] $
 +
and $  [ - \rho , 0 ] $(
 +
Krein's criterion). The system  $  \{ \psi _ {j} \} $
 +
of eigen vectors corresponding to different eigen values  $  \lambda _ {j} $,
 +
$  j = 1 , 2 \dots $
 +
of a dissipative operator forms a basis of its closed linear span and is equivalent to an orthonormal basis if
  
<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/d/d033/d033430/d03343058.png" /></td> </tr></table>
+
$$
 +
\sum _ {\begin{array}{c}
 +
j , k = 1 , \\
 +
j \neq k
 +
\end{array}
 +
} ^  \infty 
 +
 
 +
\frac{ \mathop{\rm Im}  \lambda _ {j}  \mathop{\rm Im}  \lambda _ {k} }{| \lambda _ {j} - \overline \lambda \; _ {k} | }
 +
  < \infty .
 +
$$
 +
 
 +
The concept of a dissipative operator was also introduced for non-linear and even for multi-valued operators  $  A $.  
 +
Such an operator on a Hilbert space is called dissipative if for any two of its values the inequality
 +
 
 +
$$
 +
\mathop{\rm Re}  ( A x _ {1} - A x _ {2} , x _ {1} - x _ {2} )  \leq  0
 +
$$
  
 
holds. This concept also forms the base of the theory of one-parameter non-linear contraction semi-groups and the related differential equations. Another generalization of the concept of a dissipative operator concerns operators acting on a Banach space with a so-called semi-inner product. Another generalization concerns operators acting on a [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]].
 
holds. This concept also forms the base of the theory of one-parameter non-linear contraction semi-groups and the related differential equations. Another generalization of the concept of a dissipative operator concerns operators acting on a Banach space with a so-called semi-inner product. Another generalization concerns operators acting on a [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]].
Line 41: Line 132:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.S. Livshits,  "On the spectral resolution of linear non-selfadjoint operators"  ''Transl. Amer. Math. Soc. (2)'' , '''5'''  (1957)  pp. 67–114  ''Mat. Sb.'' , '''34''' :  1  (1954)  pp. 145–199</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.S. Phillips,  "Dissipative operators and hyperbolic systems of partial differential equations"  ''Trans. Amer. Math. Soc.'' , '''90''' :  2  (1959)  pp. 193–254</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Crandall,  A. Pazy,  "Semi-groups of nonlinear contractions"  ''J. Funct. Anal.'' , '''3'''  (1969)  pp. 376–418</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G. Lumer,  R. Phillips,  "Dissipative operators in a Banach space"  ''Pacific J. Math.'' , '''11'''  (1961)  pp. 679–698</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.S. Livshits,  "On the spectral resolution of linear non-selfadjoint operators"  ''Transl. Amer. Math. Soc. (2)'' , '''5'''  (1957)  pp. 67–114  ''Mat. Sb.'' , '''34''' :  1  (1954)  pp. 145–199</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.S. Phillips,  "Dissipative operators and hyperbolic systems of partial differential equations"  ''Trans. Amer. Math. Soc.'' , '''90''' :  2  (1959)  pp. 193–254</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Crandall,  A. Pazy,  "Semi-groups of nonlinear contractions"  ''J. Funct. Anal.'' , '''3'''  (1969)  pp. 376–418</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G. Lumer,  R. Phillips,  "Dissipative operators in a Banach space"  ''Pacific J. Math.'' , '''11'''  (1961)  pp. 679–698</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:36, 5 June 2020


A linear operator $ A $ defined on a domain $ D _ {A} $ which is dense in a Hilbert space $ H $ such that

$$ \mathop{\rm Im} ( A x , x ) \geq 0 \ \textrm{ if } x \in D _ {A} . $$

This requirement is sometimes replaced by the condition $ \mathop{\rm Re} ( A x , x ) \leq 0 $ if $ x \in D _ {A} $, i.e. the dissipativeness of $ A $ in this sense is equivalent to that of the operator $ ( - iA ) $.

A dissipative operator is said to be maximal if it has no proper dissipative extensions. A dissipative operator always has a closure, which also is a dissipative operator; in particular, a maximal dissipative operator is a closed operator. Any dissipative operator can be extended to a maximal dissipative operator. For a dissipative operator all points $ \lambda $ with $ \mathop{\rm Im} \lambda < 0 $ belong to the resolvent set, and moreover

$$ \| A x - \lambda x \| \geq | \mathop{\rm Im} \lambda | \| x \| , \ x \in D _ {A} . $$

A dissipative operator is maximal if and only if $ ( A - \lambda I ) D _ {A} = H $ for all $ \lambda $ with $ \mathop{\rm Im} \lambda < 0 $. An equivalent condition for maximality of a dissipative operator is that it is closed and that

$$ \mathop{\rm Im} ( A ^ {*} y , y ) \leq 0 ,\ y \in D _ {A ^ {*} } . $$

If $ A _ {0} $ is a maximal symmetric operator, then either $ A _ {0} $ or $ ( - A _ {0} ) $ is a maximal dissipative operator. Dissipative and, in particular, maximal dissipative extensions may be considered for an arbitrary symmetric operator $ A _ {0} $; their description is equivalent to the description of all maximal dissipative extensions of the conservative operator $ B _ {0} = iA _ {0} $: $ \mathop{\rm Re} ( B _ {0} x , x ) = 0 $, $ x \in D _ {B} $.

Dissipative operators are closely connected with contractions (cf. Contraction) and with the so-called accretive operators, i.e. operators $ A $ for which $ iA $ is a dissipative operator. In particular, an accretive operator $ A $ is maximal if and only if $ ( - A ) $ is the generating operator (or generator) of a continuous one-parameter contraction semi-group $ \{ T _ {s} \} _ {s \geq 0 } $ on $ H $. The Cayley transform

$$ T = ( A - I ) ( A + I ) ^ {-} 1 ,\ \ A = ( I + T ) ( I - T ) ^ {-} 1 , $$

where $ A $ is a maximal accretive operator and $ T $ is a contraction not having $ \lambda = 1 $ as an eigen value, is used to construct the functional calculus and, in particular, the theory of fractional powers of maximal dissipative operators.

In the case of bounded linear operators $ A $ the definition of a dissipative operator is equivalent to the requirement $ A _ {J} \geq 0 $, where $ A _ {J} = ( A ^ {*} - A ) / 2 i $ is the imaginary part of the operator $ A $. For a completely-continuous dissipative operator $ A $ on a separable Hilbert space $ H $ with nuclear imaginary part $ A _ {J} $, several criteria (i.e. necessary and sufficient conditions) for the completeness of the system of its root vectors are available; for example,

$$ \sum _ {j = 1 } ^ { \nu ( A) } \mathop{\rm Im} \lambda _ {j} ( A) = \ \mathop{\rm tr} A _ {J} , $$

where $ \lambda _ {j} ( A) $ are all eigen values of the operator $ A $, $ j = 1 \dots \nu ( A) \leq \infty $, and $ \mathop{\rm tr} A _ {J} $ is the trace of the operator $ A _ {J} $( Livshits' criterion);

$$ \lim\limits _ {\rho \rightarrow \infty } \frac{n _ {+} ( \rho , A _ {R} ) } \rho = 0 \ \textrm{ or } \ \lim\limits _ {\rho \rightarrow \infty } \frac{n _ {-} ( \rho , A _ {R} ) } \rho = 0 , $$

where $ A _ {R} = ( A + A ^ {*} ) / 2 $ is the real part of $ A $, and $ n _ \pm $ is the number of characteristic numbers of the operator $ A _ {R} $ in the segment $ [ 0 , \rho ] $ and $ [ - \rho , 0 ] $( Krein's criterion). The system $ \{ \psi _ {j} \} $ of eigen vectors corresponding to different eigen values $ \lambda _ {j} $, $ j = 1 , 2 \dots $ of a dissipative operator forms a basis of its closed linear span and is equivalent to an orthonormal basis if

$$ \sum _ {\begin{array}{c} j , k = 1 , \\ j \neq k \end{array} } ^ \infty \frac{ \mathop{\rm Im} \lambda _ {j} \mathop{\rm Im} \lambda _ {k} }{| \lambda _ {j} - \overline \lambda \; _ {k} | } < \infty . $$

The concept of a dissipative operator was also introduced for non-linear and even for multi-valued operators $ A $. Such an operator on a Hilbert space is called dissipative if for any two of its values the inequality

$$ \mathop{\rm Re} ( A x _ {1} - A x _ {2} , x _ {1} - x _ {2} ) \leq 0 $$

holds. This concept also forms the base of the theory of one-parameter non-linear contraction semi-groups and the related differential equations. Another generalization of the concept of a dissipative operator concerns operators acting on a Banach space with a so-called semi-inner product. Another generalization concerns operators acting on a Hilbert space with an indefinite metric.

References

[1] B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)
[2] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)
[3] M.S. Livshits, "On the spectral resolution of linear non-selfadjoint operators" Transl. Amer. Math. Soc. (2) , 5 (1957) pp. 67–114 Mat. Sb. , 34 : 1 (1954) pp. 145–199
[4] R.S. Phillips, "Dissipative operators and hyperbolic systems of partial differential equations" Trans. Amer. Math. Soc. , 90 : 2 (1959) pp. 193–254
[5] M. Crandall, A. Pazy, "Semi-groups of nonlinear contractions" J. Funct. Anal. , 3 (1969) pp. 376–418
[6] G. Lumer, R. Phillips, "Dissipative operators in a Banach space" Pacific J. Math. , 11 (1961) pp. 679–698

Comments

A good reference for dissipative operators on more general spaces than Hilbert spaces is [a1]. For operators on Hilbert spaces see also [a2].

References

[a1] H.O. Fattorini, "The Cauchy problem" , Addison-Wesley (1983) pp. 120–125; 154–159
[a2] I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Dissipative operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dissipative_operator&oldid=46750
This article was adapted from an original article by I.S. Iokhvidov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article