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A family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681902.png" />, acting in a Banach or topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681903.png" />, with the property
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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/o/o068/o068190/o0681904.png" /></td> </tr></table>
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If the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681905.png" /> are linear, bounded and are acting in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681906.png" />, then the measurability of all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681908.png" />, implies their continuity. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o0681909.png" /> increases no faster than exponentially at infinity. The classification of one-parameter semi-groups is based on their behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o06819010.png" />. In the simplest case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o06819011.png" /> is strongly convergent to the identity operator as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o06819012.png" /> (see [[Semi-group of operators|Semi-group of operators]]).
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A family of operators  $  T ( t) $,
 +
$  t > 0 $,
 +
acting in a Banach or topological vector space  $  X $,
 +
with the property
 +
 
 +
$$
 +
T ( t + \tau ) x  = \
 +
T ( t) [ T ( \tau ) x],\ \
 +
t, \tau > 0,\ \
 +
x \in X.
 +
$$
 +
 
 +
If the operators  $  T ( t) $
 +
are linear, bounded and are acting in a Banach space $  X $,  
 +
then the measurability of all the functions $  T ( t) x $,  
 +
$  x \in X $,  
 +
implies their continuity. The function $  \| T ( t) \| $
 +
increases no faster than exponentially at infinity. The classification of one-parameter semi-groups is based on their behaviour as $  t \rightarrow 0 $.  
 +
In the simplest case $  T ( t) $
 +
is strongly convergent to the identity operator as $  t \rightarrow 0 $(
 +
see [[Semi-group of operators|Semi-group of operators]]).
  
 
An important characteristic of a one-parameter semi-group is the [[Generating operator of a semi-group|generating operator of a semi-group]]. The basic problem in the theory of one-parameter semi-groups is the establishment of relations between properties of semi-groups and their generating operators. One-parameter semi-groups of continuous linear operators in locally convex spaces have been studied rather completely.
 
An important characteristic of a one-parameter semi-group is the [[Generating operator of a semi-group|generating operator of a semi-group]]. The basic problem in the theory of one-parameter semi-groups is the establishment of relations between properties of semi-groups and their generating operators. One-parameter semi-groups of continuous linear operators in locally convex spaces have been studied rather completely.
  
One-parameter semi-groups of non-linear operators in Banach spaces have been investigated in the case when the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068190/o06819013.png" /> are contractive. There are deep connections here with the theory of dissipative operators.
+
One-parameter semi-groups of non-linear operators in Banach spaces have been investigated in the case when the operators $  T ( t) $
 +
are contractive. There are deep connections here with the theory of dissipative operators.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1  {{MR|0617913}} {{ZBL|0435.46002}} </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)  {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)  {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semigroups of operators and approximation" , Springer  (1967)  {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)  {{MR|0390843}} {{ZBL|0328.47035}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Davies,  "One-parameter semigroups" , Acad. Press  (1980)  {{MR|0591851}} {{ZBL|0457.47030}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Goldstein,  "Semigroups of linear operators and applications" , Oxford Univ. Press  (1985)  {{MR|0790497}} {{ZBL|0592.47034}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1  {{MR|0617913}} {{ZBL|0435.46002}} </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)  {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)  {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semigroups of operators and approximation" , Springer  (1967)  {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)  {{MR|0390843}} {{ZBL|0328.47035}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Davies,  "One-parameter semigroups" , Acad. Press  (1980)  {{MR|0591851}} {{ZBL|0457.47030}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Goldstein,  "Semigroups of linear operators and applications" , Oxford Univ. Press  (1985)  {{MR|0790497}} {{ZBL|0592.47034}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)  {{MR|0710486}} {{ZBL|0516.47023}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ph. Clément,  H.J.A.M. Heijmans,  S. Angenent,  C.J. van Duijn,  B. de Pagter,  "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland  (1987)  {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Brezis,  "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)  {{MR|348562}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Casteren,  "Generators of strongly continuous semigroups" , Pitman  (1985)  {{MR|}} {{ZBL|0576.47023}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Nagel (ed.) , ''One-parameter semigroups of positive operators'' , Springer  (1986)  {{MR|0839450}} {{ZBL|0585.47030}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)  {{MR|0710486}} {{ZBL|0516.47023}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ph. Clément,  H.J.A.M. Heijmans,  S. Angenent,  C.J. van Duijn,  B. de Pagter,  "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland  (1987)  {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Brezis,  "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)  {{MR|348562}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Casteren,  "Generators of strongly continuous semigroups" , Pitman  (1985)  {{MR|}} {{ZBL|0576.47023}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Nagel (ed.) , ''One-parameter semigroups of positive operators'' , Springer  (1986)  {{MR|0839450}} {{ZBL|0585.47030}} </TD></TR></table>

Latest revision as of 08:04, 6 June 2020


A family of operators $ T ( t) $, $ t > 0 $, acting in a Banach or topological vector space $ X $, with the property

$$ T ( t + \tau ) x = \ T ( t) [ T ( \tau ) x],\ \ t, \tau > 0,\ \ x \in X. $$

If the operators $ T ( t) $ are linear, bounded and are acting in a Banach space $ X $, then the measurability of all the functions $ T ( t) x $, $ x \in X $, implies their continuity. The function $ \| T ( t) \| $ increases no faster than exponentially at infinity. The classification of one-parameter semi-groups is based on their behaviour as $ t \rightarrow 0 $. In the simplest case $ T ( t) $ is strongly convergent to the identity operator as $ t \rightarrow 0 $( see Semi-group of operators).

An important characteristic of a one-parameter semi-group is the generating operator of a semi-group. The basic problem in the theory of one-parameter semi-groups is the establishment of relations between properties of semi-groups and their generating operators. One-parameter semi-groups of continuous linear operators in locally convex spaces have been studied rather completely.

One-parameter semi-groups of non-linear operators in Banach spaces have been investigated in the case when the operators $ T ( t) $ are contractive. There are deep connections here with the theory of dissipative operators.

References

[1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 MR0617913 Zbl 0435.46002
[2] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701
[3] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501
[4] P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) MR230022
[5] V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian) MR0390843 Zbl 0328.47035
[6] E.B. Davies, "One-parameter semigroups" , Acad. Press (1980) MR0591851 Zbl 0457.47030
[7] J.A. Goldstein, "Semigroups of linear operators and applications" , Oxford Univ. Press (1985) MR0790497 Zbl 0592.47034

Comments

References

[a1] A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) MR0710486 Zbl 0516.47023
[a2] Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987) MR0915552 Zbl 0636.47051
[a3] H. Brezis, "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973) MR348562
[a4] J. van Casteren, "Generators of strongly continuous semigroups" , Pitman (1985) Zbl 0576.47023
[a5] R. Nagel (ed.) , One-parameter semigroups of positive operators , Springer (1986) MR0839450 Zbl 0585.47030
How to Cite This Entry:
One-parameter semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_semi-group&oldid=28256
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article