Alternative Infinitesimal Generator of Invertible Evolution Families

A logarithm representation of evolution operators is defined. Generators of invertible evolution families are characterized by the logarithm representation. In this article, using the logarithm representation, a concept of evolution operators without satisfying the semigroup property is introduced. In conclusion the existence of alternative infinitesimal generator is clarified.


Introduction
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 [1]), 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. ∫ 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 ∫ 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 [9].In this article the logarithm representation of evolution operator is shown to lead to the concept of evolution operator without satisfying the semigroup property.

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 > 0 T , let a certain time interval [ ] family of two-parameter evolution operator on X be

U t r U r s U t s U s s I
The property (1) is called the semigroup property.In addition the inverse operator is assumed to exist: Indeed the 0 C -group property can be confirmed by ,0 .
t s u r t s r r u r U t s r r U r u U t s r u u t s r Consequently the 0 C -group ( 0 C -semigroup) property of ( ) from the definition of ( ) According to the standard theory of linear evolution equations [1], the following boundedness is assumed; there exists real numbers M and β such that , 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.

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 u Y X ∈ ⊂ .Since there is an arbitrariness of choosing the dense subspace of X, Y can be different depending on the detail of ( ) , U t s .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 t ∂ , follows as follows.This is a linear evolution equation of autonomous type.In this manner the pre-infinitesimal generator of ( ) , U t s is obtained as the operator A(t) satisfying Equation (3).

Function of Operator
It is sufficient to consider the function of bounded operators, since ( ) , U t s is bounded on X.As a framework of defining functions of bounded operator, the Dunford-Riesz integral [10] ( ) ( ) 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 Γ consists of Jordan curves including all the spectral sets of ( ) • the integral path Γ 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.

Logarithmic Function of Operator
The logarithm of ( ) , U t s is defined using the Dunford-Riesz integral.Let the principal branch of logarithm be denoted by Log .For a certain complex number 0 κ ≠ , the logarithm of ( )

∫
where I κ plays a role of moving the spectral set of ( ) between the infinitesimal generator and the logarithm of operator has been proved in Ref. [9].Let us introduce a notation: Since ( ) , a t s is bounded on X, it is possible to define the exponential function of ( )

Evolution Operator without Satisfying the Semigroup Property
If ( ) This means that a group ( ) , U t s is generated by possibly unbounded opera- tors 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 ( ) , a t s with a certain complex number κ : does not satisfy the semigroup property:

Main Result
Introduction of 0 κ ≠ is the key to obtain the logarithmic representation, as well as to find the operator ( )   , e a t s .Indeed it is always possible to define ( )   , e a t s for a certain κ ∈ » .As seen in the preceding discussion, the singularity treatment de- pends on the boundedness property, which results from the finiteness of the in- Theorem 1.For the operator ( )   , e a t s on X, the semigroup property is replaced with a t s a t r a r s a t r a r s a s s The inverse relation is replaced with a s t a t s a s s a t s a s t I κ κ κ In particular the commutation a t r a r s a t s where, by taking κ with a large κ , κ is possible to be taken as common to ( ) , U t s with different t and s.Meanwhile the replacement of  Equations ( 12) and ( 13) show the commutativity and violation of semigroup property by ( )   , e a t s .The right hand sides of Equations ( 12) and ( 13) are equal to zero for 0 κ = .These situations correspond to the cases when the semigroup property is satisfied by ( )   , e a t s , and we see that the insufficiency of semigroup property is ultimately reduced to the introduction of nonzero κ .
The decomposition is obtained by the following constitution theorem for the evolution operator.Note that the decomposition of ( ) , e a t s also provides a certain relation between the time-discretization and the violation of semigroup property.
Theorem 2. For a given decomposition { } ( ) ( ) ( ) where 0 r and are the solutions of Equation (7) with different coefficients.
Proof.According to Equation (12), a decomposition 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 where an initial value 0 u 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 , and assumed in the standard theory of evolution equations.Here, under the existence of with focusing on the replacement of the original semigroup property.
is assumed, it is necessarily possible to take an appro- priate integral path Γ by adjusting the amplitude of κ .If clude the origin.In addition, according to the preceding discussion, it is necessary for an integral path Γ to include the spectral set of ( ) a t s by a convergent power series.Meanwhile a t s with different t and s are further assumed to commute, the exponens , it can be arbitrarily taken from X. Therefore, if we take u