Semi-group of operators

A family of operators on a Banach space or topological vector space with the property that the composite of any two operators in the family is again a member of the family. If the operators are "indexed" by elements of some abstract semi-group and the binary operation of the latter is compatible with the composition of operators, is known as a representation of the semi-group . The most detailed attention has been given to one-parameter semi-groups (cf. One-parameter semi-group) of bounded linear operators on a Banach space , which yield a representation of the additive semi-group of all positive real numbers, i.e. families with the property

If is strongly measurable, , then is a strongly-continuous semi-group; this will be assumed in the sequel.

The limit

exists; it is known as the type of the semi-group. The functions increase at most exponentially.

An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:

defined on the linear set of all elements for which the limit exists; the closure, , of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let be the subspace defined as the closure of the union of all values ; then is dense in . If there are no non-zero elements in such that , then the generating operator exists. In the sequel it will be assumed that and that implies .

The simplest class of semi-groups, denoted by , is defined by the condition: as for any . This is equivalent to the condition: The function is bounded on any interval . In that case has a generating operator whose resolvent satisfies the inequalities

(1)

where is the type of the semi-group. Conversely, if is a closed operator with domain of definition dense in and with a resolvent satisfying (1), then it is the generating operator of some semi-group of class such that . Condition (1) is satisfied if

(the Hill–Yosida condition). If, moreover, , then is a contraction semi-group: .

A summable semi-group is a semi-group for which the functions are summable on any finite interval for all . A summable semi-group has a generating operator . The operator is closed if and only if, for every ,

For one can define the Laplace transform of a summable semi-group,

(2)

giving a bounded linear operator which has many properties of a resolvent operator.

A closed operator with domain of definition dense in is the generating operator of a summable semi-group if and only if, for some , the resolvent exists for and the following conditions hold: a) , ; b) there exist a non-negative function , , , jointly continuous in all its variables, and a non-negative function , bounded on any interval , such that, for ,

Under these conditions

If one requires in addition that the function be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function such that, for ,

(3)

(4)

Under these conditions, . By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If , the result is the class and (1) follows from (4). If , , condition (4) implies the condition

Semi-groups with power singularities.

If in the previous example , then the integrals in (4) are divergent for . Hence the generating operator for the corresponding semi-group may not have a resolvent for any , i.e. it may have a spectrum equal to the entire complex plane. However, for large enough one can define for such operators functions which coincide with the functions in the previous cases. The operator function is called a resolvent of order if it is analytic in some domain and if for ,

and if for all implies . If , the operator may have a unique resolvent of order , for which there is a maximal domain of analyticity, known as the resolvent set of order .

Let be a strongly-continuous semi-group such that the inequality

holds for . Then its generating operator has a resolvent of order for , and, moreover,

(5)

Conversely, suppose that for the operator has a resolvent of order satisfying (5) with . Then there exists a unique semi-group such that

and the generating operator of this semi-group is such that .

Smooth semi-groups.

If , the function is continuously differentiable and

There exist semi-groups of class such that, if , the functions are non-differentiable for all . However, there are important classes of semi-groups for which the degree of smoothness increases with increasing . If the functions , , are differentiable for any , then it follows from the semi-group property that the are twice differentiable if , three times differentiable if , etc. Therefore, if these functions are differentiable at any for , then is infinitely differentiable.

Given a semi-group of class , a necessary and sufficient condition for the functions to be differentiable for all and , where , is that there exist numbers such that the resolvent is defined in the domain

while in this domain

A necessary and sufficient condition for to be infinitely differentiable for all and is that, for every , there exist such that the resolvent is defined in the domain

and such that

Sufficient conditions are: If there exists a for which

then the are differentiable for and ; if , then the are infinitely differentiable for all and .

The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every there exists a such that, for ,

then the are infinitely differentiable for all , .

There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree and is infinitely differentiable for , then the function

also has polynomial growth:

In the general case there is no rigorous relationship between the numbers and , and can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.

Analytic semi-groups.

An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class has this property if and only if its resolvent satisfies the following inequality in some right half-plane :

Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate

Finally, the inequality

is also a sufficient condition for to be analytic.

If a semi-group has an analytic continuation to a sector and has polynomial growth at zero, , , then the resolvent of order of its generating operator has an analytic continuation to the sector , and satisfies the following estimate in any sector , :

Conversely, suppose that the resolvent of an operator is defined in a sector and that

Then there exists a semi-group of growth , analytic in the sector , whose generating operator is such that .

Distribution semi-groups.

In accordance with the general concept of the theory of distributions (cf. Generalized function), one can drop the requirement that the operator-valued function be defined for every , demanding only that it be possible to evaluate the integrals for all in the space of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space is a continuous linear mapping of into the space of all bounded linear operators on , with the following properties: a) if ; b) if are functions in the subspace of all functions in with support in , then , where the star denotes convolution:

(the semi-group property); c) if for all , then ; d) the linear hull of the set of all values of , , , is dense in ; e) for any , , there exists a continuous on with values in , so that and

for all .

The infinitesimal operator of a distribution semi-group is defined as follows. If there exists a delta-sequence such that and as , then and . The infinitesimal operator has a closure , known as the infinitesimal generator of the distribution semi-group. The set is dense in and contains for any .

A closed linear operator with a dense domain of definition in is the infinitesimal generator of a distribution semi-group if and only if there exist numbers , and a natural number such that the resolvent exists for and satisfies the inequality

(6)

If is a closed linear operator on , then the set can be made into a Fréchet space by introducing the system of norms

The restriction of to leaves invariant. If is the infinitesimal generator of a semi-group, then is the infinitesimal generator of a semi-group of class (continuous for , ) on . Conversely, if is dense in , the operator has a non-empty resolvent set and is the infinitesimal generator of a semi-group of class on , then is the infinitesimal generator of a distribution semi-group on .

A distribution semi-group has exponential growth of order at most , , if there exists an such that is a continuous mapping in the topology induced on by the space of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent which satisfies (6) in the domain

where . In particular, if the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator . Questions of smoothness and analyticity have also been investigated for distribution semi-groups.

Semi-groups of operators in a (separable) locally convex space .

The definition of a strongly-continuous semi-group of operators continuous on remains the same as for a Banach space. Similarly, the class is defined by the property as for any . A semi-group is said to be locally equicontinuous (of class ) if the family of operators is equicontinuous when ranges over any finite interval in . In a barrelled space, a semi-group of class is always equicontinuous (cf. Equicontinuity).

A semi-group is said to be equicontinuous (of class ) if the family , , is equicontinuous.

Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.

Assume from now on that the space is sequentially complete. The infinitesimal generator of a semi-group of class is identical to the infinitesimal operator; its domain of definition, , is dense in and, moreover, the set is dense in . The semi-group leaves invariant and

If is the infinitesimal generator of a semi-group of class , the resolvent is defined for and is the Laplace transform of the semi-group.

A linear operator is the infinitesimal generator of a semi-group of class if and only if it is closed, has dense domain of definition in , and if there exists a sequence of positive numbers such that, for any , the resolvent is defined and the family of operators , is equicontinuous. In this situation the semi-group can be constructed by the formula

In a non-normed locally convex space, the infinitesimal generator of a semi-group of class may have no resolvent at any point. An example is: in the space of infinitely-differentiable functions of on . As a substitute for the resolvent one can take a continuous operator whose product with , from the right and the left, differs by a "small amount" from the identity operator.

A continuous operator defined for in a set is called an asymptotic resolvent for a linear operator if is continuous on , the operator can be extended from to a continuous operator on , and if there exists a limit point of the set such that , as for any , where

An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.

A closed linear operator with a dense domain of definition in is the infinitesimal generator of a semi-group of class if and only if there exist numbers and such that, for , there exists an asymptotic resolvent of with the properties: the functions , , are strongly infinitely differentiable for , and the families of operators

are equicontinuous.

Generation theorems have also been proved for other classes of semi-groups of operators on a locally convex space.

Adjoint semi-groups.

If is a semi-group of class on a Banach space , then the adjoint operators form a semi-group of bounded operators on the adjoint space . However, the assertion that as for any is valid only in the sense of the weak- topology . If is the generating operator, its adjoint is a weak infinitesimal generator for , in the sense that is the set of all for which the limit of as exists in the sense of weak- convergence and is equal to . The domain of definition is dense in — again in the sense of the weak- topology — and the operator is closed in the weak- topology.

Let be the set of all elements in such that as in the strong sense; then is a closed subspace of that is invariant under all . On the operators form a semi-group of class . The space is also the strong closure of the set in . If the original space is reflexive, then . Analogous propositions hold for semi-groups of class in locally convex spaces. Semi-groups of classes and generate semi-groups of the same classes in .

Distribution semi-groups in a (separable) locally convex space.

A distribution semi-group in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group is said to be locally equicontinuous (of class ) if, for any compact subset , the family of operators , , is equicontinuous. In a barrelled space , any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class , the infinitesimal operator is closed , is dense in , and for any and ,

(7)

A generalized function with support in , possessing the properties (7), is naturally called the fundamental function of the operator . Thus, if is the infinitesimal operator of a semi-group of class , then is the fundamental function of the operator . The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function (or, more precisely, of the function , where ).

A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let denote the Laplace transform of a function , and let be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of . The Laplace transform of an -valued generalized function is defined by . Under these conditions, is a continuous mapping of into the space of continuous linear operators on . Let be the space of all obtained from functions with support in , with the natural topology. If is a linear operator on , it can be "lifted" to an operator on via the equality

Thus, it is defined for all such that the right-hand side of the equality is defined for any and it extends to a generalized function in . The continuous operator on is defined by

If the operator has a continuous inverse on , then is called the generalized resolvent of .

An operator has a generalized resolvent if and only if the operator has a locally equicontinuous fundamental function , constructed by the formula

where

Subject to certain additional assumptions, is a distribution semi-group. An extension theorem for semi-groups of class has also been proved in terms of generalized resolvents.

See also Semi-group of non-linear operators.

References

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