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.
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One-parameter semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_semi-group&oldid=48041