# One-parameter semi-group

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.

How to Cite This Entry:
One-parameter semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_semi-group&oldid=48041
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article