Generating operator of a semi-group

The derivative at zero of a semi-group of linear operators (cf. Semi-group of operators) , , acting on a complex Banach space . If is continuous in the operator norm, then it has the form , where is a bounded operator,

(1)

for any , and is the generating operator of this semi-group. Conversely, if the limit on the left-hand side exists, for all , then .

A more complicated situation arises when is only a strongly-continuous semi-group. In this case the limit (1) does not necessarily exist for every . The operator , defined on the linear set of all for which the limit exists, is linear and unbounded, and is called the infinitesimal operator. In particular, is defined on all elements of the form , , . If denotes the closure of the union of the range of values of all , , then is dense in and, moreover, is dense in . The values of also lie in . If is an unbounded operator, then is a set of the first category in .

If does not contain elements for which , then has a closure , which is also called the generating operator of the semi-group . In this case, for ,

(2)

These equations define an operator which is, generally speaking, an extension of the closure of . It is also called the generalized generating operator of .

On the set of all for which the improper integral

(3)

converges, one defines the operator

for , where is the type of the semi-group . This operator has the following properties:

1) ;

2) ;

3) , ;

4) , .

If the integral (3) converges absolutely for any , then the generating operator exists if and only if , , implies ; the operator is bounded and, if , it coincides with the resolvent of . For to be closed (i.e. for ) it is necessary and sufficient that

for any .

The basic problem in the theory of operator semi-groups is to establish relations between properties of a semi-group and properties of its generating operator, where the latter are usually formulated in terms of .

References

[1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2]	P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" Soviet Math. Dokl. , 10 : 6 (1969) pp. 1523–1526 Dokl. Akad. Nauk , 189 : 5 (1969) pp. 934–937
[3]	A.V. Zafievskii, "On semigroups with singularities summable with a power-weight at zero" Soviet Math. Dokl. , 11 : 6 (1970) pp. 1408–1411 Dokl. Akad. Nauk , 195 : 1 (1970) pp. 24–27

References

[a1]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)
[a2]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)