For a given evolution operator let its logarithm function be well-defined. A simple question arises here; “is there any difference between the logarithm of evolution operator and the infinitesimal generator?” This question is associated with the unboundedness of infinitesimal generator. However a role of the unboundedness of the infinitesimal generator has not been understood well so far. Indeed, in the standard theory of linear evolution equation (for example, see ), the evolution operator is treated as a bounded operator in a given functional space X regardless of whether the infinitesimal generator is bounded or unbounded in X.
This question is considered in a concrete framework of abstract Cauchy problem. Partial differential equations are regarded as ordinary differential equations in functional spaces. The initial value problems of autonomous evolution equations are written by
in X, where an initial value is given in X, and A(t) is generally unbounded in X. If there is a solution for this initial value problem, its solution is formally represented by
under the well-definedness of the indefinite integral and its exponential function. The evolution operators appearing in the following discussion correspond to the above exponential function. Note that, since A(t) is generally given as an unbounded operator in X, the exponential of A(t) is not necessarily well-defined even if A(t) is independent of t (cf. Hille-Yosida theorem).
There is a long history of studying logarithm of operators -. The logarithm of is defined under a certain setting and such a logarithm is clarified to play a role of extracting an essential and bounded part of infinitesimal generator . In this article the logarithm representation of evolution operator is shown to lead to the concept of evolution operator without satisfying the semigroup property.
2. Evolution Operator and Its Infinitesimal Generator
2.1. Invertible Evolution Operator
An evolution operator is assumed to be defined on a Banach space X. Although evolution operator is not necessarily invertible, here we confine ourselves to invertible evolution operators. The reason for this restriction can be found in Sec. 3.2.
For , let a certain time interval satisfy . Let a
family of two-parameter evolution operator on X be satisfying
The property (1) is called the semigroup property. In addition the inverse operator is assumed to exist:
If is generated by the operator independent of t (and s), one-para- meter group can be defined by
utilizing this two-parameter evolution operator. Since is a real number only satisfying , it can be negative. If a solution at certain times t and s are represented by and respectively, it is trivially seen from the definition that is a mapping from to . is nothing but a -group generated by t-independent infinitesimal generator, and the following properties are satisfied:
Indeed the -group property can be confirmed by
Furthermore, for , application of to leads to
Consequently the -group ( -semigroup) property of is derived from the definition of .
According to the standard theory of linear evolution equations , the following boundedness is assumed; there exists real numbers M and such that
for , where denotes an operator norm. This assumption restricts the time evolution to be linearly bounded. Note that, using the equality , the assumption can be replaced with
without the essential difference.
2.2. Pre-Infinitesimal Generator
The infinitesimal generator is defined using the evolution operator. Let the dense subspace Y of X be non-empty space admitting the definition of the following weak limit:
for . Since there is an arbitrariness of choosing the dense subspace of X, Y can be different depending on the detail of . Under the existence of the above weak limit, the infinitesimal generator is defined by
Since A(t) is defined under a weaker assumption compared to the standard theory of evolution equations, we call this operator the pre-infinitesimal generator. The definition of weak t-differential, which is denoted as , follows as
where the relation is used. In this article we consider the subspace Y as a natural choice of domain space . Generally speaking, if A(t) is dependent on t, does depend on t. Here, by considering sufficiently small interval , is assumed to be independent of t and s. Furthermore, by taking ,
follows. This is a linear evolution equation of autonomous type. In this manner the pre-infinitesimal generator of is obtained as the operator A(t) satisfying Equation (3).
3. Logarithmic Representation of Infinitesimal Generator
3.1. Function of Operator
It is sufficient to consider the function of bounded operators, since is bounded on X. As a framework of defining functions of bounded operator, the Dunford-Riesz integral 
is utilized. Note that functions of bounded operator on X are not necessarily bounded operators on X. For drawing an integral path on the complex plain,
・ the integral path consists of Jordan curves including all the spectral sets of ,
・ the integral path must not include singular points of .
That is, for the definition of logarithm of operators, it is necessary to take an integral path not to include the origin, since the origin is the singular point of logarithm function.
3.2. Logarithmic Function of Operator
The logarithm of is defined using the Dunford-Riesz integral. Let the principal branch of logarithm be denoted by . For a certain complex number , the logarithm of is defined by
where plays a role of moving the spectral set of not to include the origin. In addition, according to the preceding discussion, it is necessary for an integral path to include the spectral set of . Since the boundedness of is assumed, it is necessarily possible to take an appropriate integral path by adjusting the amplitude of . If is well-defined, then the definition of trivially follows. That is, for the present definition manner, the sign of the logarithm of operator can not be a matter. This fact is essentially arises from limiting the time interval as finite. This provides the reason why we assume as invertible.
between the infinitesimal generator and the logarithm of operator has been proved in Ref. . Let us introduce a notation:
Since is bounded on X, it is possible to define the exponential function of by a convergent power series. Meanwhile is obtained by applying the resolvent operator of to the pre-infinitesimal generator A(t).
3.3. Evolution Operator without Satisfying the Semigroup Property
If with different t and s are further assumed to commute, the exponential function satisfies
where is generally an unbounded operator in X, and it is well- defined by considering the previously-defined subspace Y. Although stands for , it can be arbitrarily taken from X. Therefore, if we take ,
is obtained, where note that s in must be common to s in . On the other hand, the exponential function of leads to
This means that a group is generated by possibly unbounded operators being represented by the convergent power series. Here it is clear that the logarithmic representation is not simply a paraphrase of Hille-Yosida theorem. Equation (8) is an alternative equation of Equation (3), where the described evolutions are not exactly the same but connected by Equation (9).
It is notable here that exponential function of with a certain complex number :
does not satisfy the semigroup property:
First of all is seen by
Next is seen by
where and satisfy
That is, the master equations of and are different, and this fact is associated with the insufficiency of semigroup property. Consequently the evolution operator without satisfying the semigroup property is clarified to be generated by .
4. Main Result
Introduction of is the key to obtain the logarithmic representation, as well as to find the operator . Indeed it is always possible to define for a certain . As seen in the preceding discussion, the singularity treatment depends on the boundedness property, which results from the finiteness of the interval in this article, and assumed in the standard theory of evo-
lution equations. Here, under the existence of , we study algebraic
properties of with focusing on the replacement of the original semigroup property.
Theorem 1. For the operator on X, the semigroup property is replaced with
The inverse relation is replaced with
In particular the commutation
is necessarily valid.
Proof. Substitution of to leads to the following relation (see Equation (11)):
where, by taking with a large , is possible to be taken as common to with different t and s. Meanwhile the replacement of with leads to the following relation (see Equation (10)):
That is, for , behaves as the unit operator. Modified version of semigroup property (i.e., (12)) has been proved. The inverse relation (13) follows readily from Equation (12). According to Equation (12),
is valid. Combination with another relation
leads to the commutation:
where is utilized.
Equations (12) and (13) show the commutativity and violation of semigroup property by . The right hand sides of Equations (12) and (13) are equal to zero for . These situations correspond to the cases when the semigroup property is satisfied by , and we see that the insufficiency of semigroup property is ultimately reduced to the introduction of nonzero .
The decomposition is obtained by the following constitution theorem for the evolution operator. Note that the decomposition of also provides a certain relation between the time-discretization and the violation of semigroup property.
Theorem 2. For a given decomposition of the interval with , the operator on X is represented by
where and in the sum are denoted as and respectively. Note that and are the solutions of Equation (7) with different coefficients.
Proof. According to Equation (12), a decomposition
is true. Another decomposition
is also true, and then
follows by sorting based on and dependence. Further decomposition shows
follows. For a certain , a constitutional representation is suggested by the deduction:
is obtained. The statement is proved by sorting terms.
Logarithm of invertible evolution families is defined by introducing nonzero . By comparing the logarithm of evolution operator to the infinitesimal generator, the difference has been found in the generated evolution operators (cf. Equation (9)). In conclusion, using the logarithmic representation, a concept of the evolution operator without satisfying the semigroup property is introduced. The violation of semigroup property has been quantitatively shown. Such an evolution operator is the alternative of original evolution operator without any loss of information.
The author is grateful to Prof. Emeritus Hiroki Tanabe for fruitful comments.
 Okazawa, N. (2000) Logarithmic Characterization of Bounded Imaginary Powers. Progress in Nonlinear Differential Equations and Their Applications, 42, 229-237. https://doi.org/10.1007/978-3-0348-8417-4_24