# Strongly-continuous semi-group

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A family , , of bounded linear operators on a Banach space with the following properties:

1) , , ;

2) the function is continuous on for any .

When 1) holds, the measurability of all functions , , and, in particular, their one-sided (right or left) weak continuity, implies strong continuity of . For a strongly-continuous semi-group the finite number

is called the type of the semi-group. Thus, the norms of the functions grow at no faster than the exponential . The classification of strongly-continuous semi-groups is based on their behaviour as . If there is a bounded operator such that as , then is a projection operator and , where is a bounded linear operator commuting with . In this case is continuous with respect to the operator norm. If , then , , is a uniformly-continuous group of operators.

If for each , then is also a projection operator, projecting onto the subspace that is the closure of the union of all , , .

For to exist and to be equal to it is necessary and sufficient that be bounded on and that . In this case the semi-group can be extended by the equality and is strongly continuous for (it satisfies the -condition). For broader classes of semi-groups the limit relation is satisfied in a generalized sense:

(Cesáro summability, the -condition), or

(Abel summability, the -condition). Here it is assumed that the function , , is integrable on (and, hence, on any finite interval).

The behaviour of a strongly-continuous semi-group as can be completely irregular. For example, the function may have a power singularity at .

For a dense set of in the function is differentiable on . An important role is played by strongly-continuous semi-groups for which the function is differentiable for all for . In this case the operator is bounded for each and its behaviour as gives new opportunities for classifying semi-groups. The classes of strongly-continuous semi-groups for which admits a holomorphic extension in a sector of the complex plane containing the semi-axis have been characterized.

#### References

 [1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)