View source for One-parameter semi-group

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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.

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====
<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====

====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>