Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.


Introduction
Based on the logarithmic representation of infinitesimal generators, a module over a Banach algebra is introduced. Let us call such an algebraic subject the BðXÞ-module, where X and BðXÞ stand for a Banach space and its operator algebra, respectively. The BðXÞ-module does not correspond only to the extension of the Banach algebra but also to the general authorization of the Lie algebra consisting of differential operators. This algebraic entity is an operator algebra being introduced based on the framework of logarithmic representation of operators. There are two concepts, which are to be bridged in this paper: a set of infinitesimal generators generating groups or semigroups of operators and the elements of the Lie algebra. The following statements are valid: (i) The sum of two closed operators are not necessarily a closed operator so that the sum of two infinitesimal generators are not necessarily an infinitesimal generator (ii) The sum of two elements in the Lie algebra are necessarily an element of the Lie algebra Here is a contradiction in some general situations, as seen in the relation between the Lie group and the Lie algebra in which the Lie algebra corresponds to a set of infinitesimal generators. Besides, these two statements are true if the two infinitesimal generators are bounded operators. More substantially, the product cannot be justified without limiting ourselves to (sub)sets of bounded operators. In this paper, by means of the logarithmic representation of infinitesimal generators of invertible evolution operators, a set of generally unbounded infinitesimal generators is characterized as an algebraic module over a Banach algebra. The logarithm of operators is a key to make a bridge for these contradicting statements.
The logarithm of an injective sectorial operator was introduced by Nollau [1] in 1969. After a long time, the logarithm of sectorial operators was studied again from 1990s [2][3][4], and its utility was established with respect to the definition of the logarithms of operators [5,6] (for a review of sectorial operators, see Hasse [7]). While the sectorial operator has been a generic framework to define the logarithm of operators, the sectorial property is not generally satisfied by the evolution operators. In this sense, it is necessary to introduce a reasonable framework for defining the logarithm of nonsectorial infinitesimal generators.
In this paper, the theory of BðXÞ nor-module is introduced. The utility of the theory is confirmed in the application to the solvability of abstract Cauchy problem, the generalization of the Cole-Hopf transform, and the foundation of the rotation group in the latter parts of Sections 2, 3, and 4. This paper is the completion of the recent studies shown in Refs. [8][9][10][11][12][13][14][15]. By organizing the preceding works into a logical order, the several statements are renewed. First, the generalized version of logarithmic representation (Corollary 8) is possible using the concept of alternative infinitesimal generator. Even without any additional assumptions, it enables us to generalize the logarithmic representation for infinitesimal generators of noninvertible evolution operators. Although this fact is taken for granted in the lately published papers of Refs. [8][9][10][11][12][13][14][15], it is mentioned within a logical process for the first time. Second, although the relativistic formulation is introduced for changing the evolution direction as seen in the application of the Cole-Hopf transform, it should not be restricted to the application of the Cole-Hopf transform. The relativistic formulation of abstract evolution equation is a kind of generalization of abstract evolution equations. More clearly, it generalizes the concept of abstract evolution equation to the abstract equation. Consequently, the theory of BðXÞ-module is written in the relativistic form.

Logarithmic Representation of Operators
2.1. Banach Algebra. Let ðA, k⋅kÞ be a Banach space (for a textbook, see [16]). A mapping is called a multiplication on A, if it is bilinear and associative. k⋅k is said to be a submultiplicative norm if kx ⋅ yk ≤ kxkkyk for each x, y ∈ A. The Banach space A together with a multiplication and submultiplicative norm is called a Banach algebra. Let X be a Banach space. Denote by BðXÞ, the set of all bounded linear operators f : X ⟶ X. Then, A = BðXÞ is an example of a Banach algebra with multiplication as composition and norm defined by where f ∈ BðXÞ. BðXÞ is called the operator algebra of X. Let A be a Banach algebra and let X be a Banach space. X is said to be As an example of the Banach algebra, BðXÞ is taken in the following. Then, X is a Banach BðXÞ-module under The Banach space, Banach algebra, and Banach BðXÞ -module (BðXÞ-module, for short) are the basic concepts in this paper.

Two
Parameter Group on Banach Spaces. All the discussion begins with the definition of groups on the Banach spaces that will be generalized to a well-defined semigroup in later sections. Let ðX, k⋅kÞ be a Banach space and BðXÞ its Banach algebra. In particular, BðXÞ is an example of a Banach algebra.
A two parameter group is defined on X. Let X be a Banach space and T ∈ ð0, ∞Þ. A two parameter group on X is an operator valued mapping ðt, sÞ ⟶ Uðt, sÞ from ½−T, T into BðXÞ with the semigroup properties: and the strong continuity; for each s ∈ ½−T, T and x ∈ X, the map t ⟶ Uðt, sÞx is continuous on ½s, T. Both Uðt, sÞ and Uðs, tÞ are assumed to be well-defined to satisfy where Uðs, tÞ corresponds to the inverse operator of Uðt, sÞ.
Since Uðt, sÞ Uðs, tÞ = Uðt, tÞ = I is also true, the commutation between Uðt, sÞ and Uðs, tÞ follows. Operator Uðt, sÞ, which is called the evolution operator in the following, is a generalization of exponential function; indeed, the properties shown in equations (7)- (9) are satisfied by taking Uðt, sÞ as e t−s . Evolution operator is an abstract concept of exponential function valid for both finite and infinite dimensional Banach spaces. Due to the validity of equation (9), the invertible evolution family is to be associated with some linear evolution equations of hyperbolic type and those of dispersive type.
In the same context, the obtained results can be directly applied to some semilinear evolution equations (for a text book, see [17]). For example, the solutions of linear and nonlinear wave equations are written by the evolution operator Uðt, sÞ defined above.

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Let Y be a dense Banach subspace of the Banach space X and the topology of Y be stronger than that of X. The space Y ⊂ X is assumed to be Uðt, sÞ-invariant; for any t, s satisfying −T ≤ t, s ≤ T, Uðt, sÞY = Y. Following the definition of C 0 -(semi) group (cf. the assumption H 2 in Section 5.3 of Pazy [18] or corresponding discussion in Kato [19,20]), Uðt, sÞ trivially satisfy the boundedness in the present setting; there exist real numbers M and β such that that are practically reduced to when the interval is restricted to be finite ½−T, T. Since C 0 semigroup theory [21][22][23], is essentially based on the Laplace transform of operators, the satisfaction of equation (10) is discussed here; Me βt in equation (10) arises from the condition for the existence theorem for the Laplace transforms (for example, see [24]), and M ′ = Me βT is regarded as a finite real number in the present setting.
Next, for the well-defined Uðt, sÞ, the counterpart of the logarithm in the abstract framework is introduced. There are two concepts associated with the logarithm of operators; one is the infinitesimal generator, and the other is t-differential of Uðt, sÞ. These two concepts are connected as follows.
Definition 1 (preinfinitesimal generator). For −T ≤ t, s ≤ T, the weak limit is assumed to exist for certain u s , which is an element of a dense subspace Y of X. A linear operator AðtÞ: Y ⟶ X is defined by for u t ∈ Y and −T ≤ t, s ≤ T. The operator AðtÞ for a whole family fUðt, sÞg −T≤t,s≤T is called the preinfinitesimal generator.
Let t-differential of Uðt, sÞ in a weak sense [13] be denoted by Equation (14) is regarded as a differential equation satisfied by uðtÞ = Uðt, sÞu s that implies a relation between AðtÞ and the logarithm: The relation between AðtÞ and the logarithm is discussed in the next Section 2.3. Preinfinitesimal generators are not necessarily infinitesimal generators without assuming a dense property of domain space Y in X. For example, in t -independent cases, an operator AðtÞ defined by equation (13) is not necessarily a densely defined and closed linear operator, while AðtÞ must be a densely defined and closed linear operator with its resolvent set included in fλ ∈ ℂ : Reλ > βg for AðtÞ to be the infinitesimal generator. On the other hand, infinitesimal generators are necessarily preinfinitesimal generators. That is, only the exponentiability with a certain ideal domain is valid to the preinfinitesimal generators. The definition of preinfinitesimal generator is useful in terms of providing the algebraic structure. Let a set of preinfinitesimal generators be denoted by GðXÞ. It is trivial that BðXÞ ⊂ GðXÞ.

Logarithmic Representation of Preinfinitesimal
Generator. The logarithmic representation of infinitesimal generator is introduced in order to clarify the structure of infinitesimal generators [8]. The logarithm of Uðt, sÞ is defined by the Dunford-Riesz integral [25]. The boundedness of Uðt, sÞ on X makes the problem rather easy. Indeed, the boundedness allows us to introduce the translation on the complex plane as a tool to realize the parallel displacement of the entire spectral set. On the other hand, two difficulties inherent to the logarithm (i) Singularity of logarithm at the origin (ii) Multivalued property of the logarithm arise. By introducing a constant κ ∈ ℂ, the singularity can be handled. This simple treatment is definitely practical to welldefine the logarithm of nonsectorial operators. By introducing a principal branch (denoted by "Log") of the logarithm (denoted by "log"), the multivalued property is handled. Indeed, for any complex number z ∈ C, a branch of logarithm is defined by where Z is a complex number chosen to satisfy |Z | = | z | , −π < arg Z ≤ π, and arg Z = arg z + 2nπ for a certain integer n.

Lemma 2.
(logarithmic representation of operators). Let t and s satisfy 0 ≤ t, s ≤ T. For a given Uðt, sÞ defined in Section 2.2, its logarithm is well defined; there exists a certain complex number κ satisfying where an integral path Γ, which excludes the origin, is a circle in the resolvent set of Uðt, sÞ + κI. Here, Γ is independent of t and s. LogðUðt, sÞ + κIÞ is bounded on X.

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Proof. The logarithm Log holds the singularity at the origin so that it is necessary to show a possibility of taking a simple closed curve (integral path) excluding the origin in order to define the logarithm by means of the Dunford-Riesz integral. It is not generally possible to take such a path in case of κ = 0: First, Uðt, sÞ is assumed to be bounded for 0 ≤ t, s ≤ T (equation (10)), and the spectral set of Uðt, sÞ is a bounded set in ℂ. Second, for κ satisfying the spectral set of Uðt, sÞ + κI is separated with the origin. Consequently, it is possible to take an integral path Γ including the spectral set of Uðt, sÞ + κI and excluding the origin. Equation (17) follows from the Dunford-Riesz integral. Furthermore, by adjusting the amplitude of κ, an appropriate integral path always exists independent of t and s. LogðUðt , sÞ + κIÞ is bounded on X, since Γ is included in the resolvent set of ðUðt, sÞ + κIÞ [Q.E.D.: Lemma 2]. ? According to this lemma, by introducing nonzero κ, the logarithm of Uðt, sÞ + κI is well-defined without assuming the sectorial property to Uðt, sÞ. On the other hand, equation (17) is valid with κ = 0 only for limited cases. Theorem 3. (logarithmic representation of infinitesimal generators). Let t and s satisfy −T ≤ t, s ≤ T and Y be a dense subspace of X. For Uðt, sÞ defined in Section 2.2, let AðtÞ ∈ GðXÞ and ∂ t Uðt, sÞ be determined by equations (13) and (14), respectively. If AðtÞ and Uðt, sÞ commute, preinfinitesimal generators fAðtÞg −T≤t≤T are represented by means of the logarithm function; there exists a certain complex number κ ≠ 0 such that where u s is an element in Y. Note that Uðt, sÞ defined in Section 2.2 is assumed to be invertible.
Proof. For Uðt, sÞ defined in Section 2.2, operators Log ðUðt , sÞ + κIÞ and Log ðUðt + h, sÞ + κIÞ are well-defined for a certain κ (Lemma 2). The t-differential in a weak sense is formally written by where Γ, which is possible to be taken independent of t, s, and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Uðt, sÞ + κI and Uðt + h, sÞ + κI. A part of the integrand of equation (20) is estimated as for v ∈ X. There are two steps to prove the validity of equation (20).
Step 1. The former part of the right hand side of equation (21) satisfies since λ is taken from the resolvent set of Uðt + h, sÞ − κI. In the same way, the operator ðλ − Uðt, sÞ − κÞ −1 is bounded on X and Y. Then, the continuity of the mapping t ⟶ ðλ − Uðt, sÞ − κÞ −1 as for the strong topology follows: Step 2. The latter part of the right hand side of equation (21) is estimated as for ∈Y. Because kAðτÞUðτ, sÞk BðY,XÞ < ∞ is true by assumption, the right hand side of equation (24) is finite. Equation (24) shows the uniform boundedness with respect to h; then, the uniform convergence ðh ⟶ 0Þ of equation (20) follows. Consequently, the weak limit process h ⟶ 0 for the integrand of equation (8) is justified, as well as the commutation between the limit and the integral.

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According to equation (20), interchange of the limit with the integral leads to for ∈Y. Because it is also allowed to interchange AðtÞ with Uðt, sÞ, for ∈Y. A part of the right hand side is calculated as due to the integration by parts, where jλ − κj = jνj = r is a properly chosen circle large enough to include Γ. ð2πiÞ −1 Ð Γ λ −1 dλ = 0 is seen by applying dLogλ/dλ = 1/λ. ð2πiÞ −1 Ð jνj=r ∑ ∞ n=1 Uðt, sÞ n ν −n−1 dν = I follows from the singularity of ν −n−1 . Consequently, is obtained for u s ∈ Y [Q.E.D.: Theorem 3]. ? The meaning of logarithmic representation is examined by focusing on ∂ t LogðUðt, sÞ + κIÞ. What is introduced by equation (19) is a kind of resolvent approximation of AðtÞ in which AðtÞ is represented by the resolvent operator of U ðs, tÞ. As seen in the following, it is notable that there is no need to take κ ⟶ 0. This point is different from the usual treatment of resolvent approximations; indeed, it is impossible to take κ ⟶ 0 if the origin is not included in the resolvent set of Uðt, sÞ. On the other hand, it is also seen by equation (19) that shows a structure of similarity transform, where ðUðt, sÞ + κ Þj κ=0 means Uðt, sÞ + κ satisfying a condition κ = 0: This asymmetric similarity transform from left and right hand sides are remarkable, and it becomes symmetric if κ = 0. A part ∂ t LogðUðt, sÞ + κIÞ plays an essential role in the following discussion.

Regularized Evolution Operator
3.1. Alternative Infinitesimal Generator. The alternative infinitesimal generator is introduced in order to extract bounded parts from the preinfinitesimal generator A [9]. The operator AðtÞ ∈ GðXÞ is generally unbounded in X. A bounded operator aðt, sÞ on X is introduced.
Definition 4 (alternative infinitesimal generator). Let κ be a certain complex number. For a certain v s ∈ X, the alternative infinitesimal generator ∂ t aðt, sÞ to AðtÞ ∈ GðXÞ is defined using on X, where ∂ t denotes t − differential in a weak sense.
In the present setting assuming the existence of ∂ t Uðt, sÞ and therefore AðtÞ, the operator ∂ t aðt, sÞ exists. According to 5 Advances in Mathematical Physics the logarithmic representation, is obtained. Since aðt, sÞ is a bounded operator defined by the Dunford-Riesz integral, v s in the definition of alternative infinitesimal generator can be taken from X. Since κ is chosen to separate the spectral set of Uðt, sÞ + κI from the origin, the inverse operator of Uðt, sÞ + κI always exists, as κ is taken from fλ ∈ ℂ ; jλj > Me βT g.
Definition 5 (regularized evolution operator). The alternative infinitesimal generator ∂ t aðt, sÞ generates the regularized evolution operator which is represented by the convergent power series.
The operator e aðt,sÞ is regularized in the following sense; the inverse evolution operator e −aðt,sÞ always exists, if e aðt,sÞ exists. This fact, which arises from the boundedness of aðt, s Þ, is true, even if negative time evolution Uðt, sÞ −1 = Uðs, tÞ is not well-defined, and only positive time evolution Uðt, sÞ is given ðt > sÞ.
It is remarkable that is always satisfied, while is not necessarily satisfied because the limited range of imaginary spectral distribution is necessarily true only for the right hand side. In this sense, aðt, sÞ corresponds to the extracted bounded part of the infinitesimal generator AðtÞ. The regularized trajectory in finite/infinite dimensional dynamical systems (for textbooks, see [26,27]) arises from the regularized evolution operator. Note that, as the welldefined ∂ t aðt, sÞ is not necessary for e aðt,sÞ to be well-defined, only the well-defined aðt, sÞ is sufficient for e aðt,sÞ to be welldefined. This fact essentially simplifies the discussion in applying e aðt,sÞ in which there is no need to consider weak differential.
Using the relation between the logarithm and the exponential functions, is valid. It shows a correspondence between e aðt,sÞ and Uðt, sÞ at the level of an evolution operator. One difference is whether the semigroup property is satisfied or not, and another difference is whether the convergence power series representation is always true or not. Meanwhile, at the level of infinitesimal generators, there is a substantial difference between aðt, sÞ and AðtÞ. That is, aðt, sÞ is always bounded on X, while AðtÞ is not necessarily bounded on X. Because of the boundedness of aðt, sÞ on X, the inverse operator e −aðt,sÞ always exists if e aðt,sÞ exists. One of the essential ideas is to generate e aðt,sÞ , instead of generating Uðt, sÞ.
Theorem 6. (modified semigroup property). Let κ be a certain complex number. For the operator e aðt,sÞ on X, the semigroup property is replaced with The inverse relation is replaced with In particular, the commutation is necessarily valid.
Proof. Substitution of Uðt, sÞ = e aðt,sÞ − κI to Uðt, rÞ Uðr, sÞ = Uðt, sÞ leads to the following relation: where by taking κ with a large |κ | , κ is possible to be taken as common to Uðt, sÞ with different t and s. Meanwhile, the replacement of Uðt, sÞ = e aðt,sÞ − κI with Uðs, sÞ = I leads to the following relation: That is, for κ ≠ 1, ðκ + 1Þ −1 e aðs:sÞ behaves as the unit operator. Modified version of semigroup property (i.e., (38)) has been proved. The inverse relation (39) follows readily from equation (38). According to equation (15),  (38) and (39) show the commutativity and violation of semigroup property by e aðt,sÞ . The right hand sides of equations (38) and (39) are equal to zero for κ = 0: These situations correspond to the cases when the semigroup property is satisfied by e aðt,sÞ , and it is readily seen that the insufficiency of semigroup property arises from the introduction of nonzero κ.
The decomposition is obtained by the following structure theorem for the regularized evolution operator. Note that the decomposition of e aðt,sÞ also provides a certain relation between the time-discretization and the violation of semigroup property.
Theorem 7. (structure of regularized evolution operator). Let κ be a certain complex number. For a given decomposition s < r 1 , r 2 , ⋯, r n < t of the interval ½s, t with n ≥ 2, the operator e aðt,sÞ on X is represented by where r 0 and r n+1 in the sum are denoted as s = r 0 and t = r n+1 , respectively.
Proof. According to equation (38), a decomposition e a t,s ð Þ = e a t,r 1 ð Þ e a r 1 ,s is true. Another decomposition e a t,r 1 ð Þ = e a t,r 2 ð Þ e a r 2 ,r 1 is also true, and then follows. For a certain n ≥ 2, a constitutional representation is suggested by the deduction: Consequently, is obtained. The statement is proved by sorting terms [Q.E.D.: Theorem 7]. ? Using the regularized evolution operator, the logarithmic representation is readily generalized to the infinitesimal 7 Advances in Mathematical Physics generators of invertible and noninvertible evolution operators. Indeed, according to the proof of Theorem 3, only the boundedness of Uðt, sÞ on X and the resulting time-interval symmetry is essential.

Corollary 8.
(generalized logarithmic representation of infinitesimal generators). Let t and s satisfy −T ≤ t, s ≤ T, and Y be a dense subspace of X. For noninvertible Uðt, sÞ, Uðt, sÞ is defined in Section 2.2 without assuming let AðtÞ ∈ GðXÞ and ∂ t Uðt, sÞ be determined by equations (13) and (14), respectively. If AðtÞ and Uðt, sÞ commute, preinfinitesimal generators fAðtÞg −T≤t≤T are represented by means of the logarithm function; there exists a certain complex number where u s is an element in Y.
Proof. The first line of equation (28) shows the validity of the statement. Indeed, for the logarithmic representation, the invertible property does not play any roles after introducing nonzero κ ∈ ℂ. In particular, ðI − κe −aðt,sÞ Þ −1 is always welldefined for a certain κ [Q.E.D.: Corollary 8]. ?
Using the regularized evolution operator, A similarity transform representation (30) for ∂ t aðt, sÞ is written by where the boundedness of aðt, sÞ allows us to define e −aðt,sÞ . Due to the boundedness of aðt, sÞ on X, e aðt,sÞ is always well-defined by a convergent power series. It leads to the holomorphic property of e aðt,sÞ . Here is the reason why e aðt,sÞ is called the regularized evolution operator. In the following, the generalized logarithmic representation of infinitesimal generators is utilized. It enables us to have the logarithmic representation not only for the C 0groups but for the C 0 -semigroups.

Renormalized Abstract Evolution Equations.
is satisfied for v s ∈ Y, where ∂ t denotes t − differential in a weak sense. This is a linear evolution equation satisfied by vð tÞ = e aðt,sÞ v s .
Proof. For the evolution operator the existence of ∂ t e aðt,sÞ is ensured by the existence of ∂ t aðt, sÞ.
Using the commutation between aðt, sÞ with different t and s, is true, and the homogeneous-type abstract evolution equation is rephrased as an equation with bounded infinitesimal generator for v s ∈ Y [Q.E.D.: Corollary 9]. ?
This is an abstract evolution equation obtained by the replacement AðtÞ with ∂ t aðt, sÞ. Here, one essential idea is to generate e aðt,sÞ instead of Uðt, sÞ; although e aðt,sÞ is easily defined due to the boundedness of aðt, sÞ, the general unboundedness of infinitesimal generator ∂ t aðt, sÞ in X is ensured by the similarity transform (56). Under the commutation assumption between AðtÞ and Uðt, sÞ, equation (60) is rephrased as where Theorem 3 is applied. Consequently, is obtained. Note again that e aðt,sÞ does not satisfy the semigroup property, while Uðt, sÞ satisfies it.

Linearized Infinitesimal
Generator. The linearity of the semigroup is not assumed in the preceding discussion so that the operator Uðt, sÞ can be taken as either linear or nonlinear semigroup. Let us assume a more general situation, in which (i) The existence of Uðt, sÞ satisfying equation (7) is locally ture for t, s ∈ ½−T, T (ii) The existence of the infinitesimal generator of Uðt, sÞ is not clear This situation corresponds to the situation when only the unique local existence of their solutions the nonlinear partial differential equations is ensured. One of the important application of the renormalized abstract evolution equation is the linearization, which enables to analyze the local profile of nonlinear semigroup. 8 Advances in Mathematical Physics

Corollary 10. (linearized evolution equation). For −T ≤ t, s
≤ T, the two parameter group is defined on X. Let either linear or nonlinear semigroup Uðt, sÞ defined on a Banach space X satisfy equation (7). Let Uðt, sÞu s be the solution of nonlinear equation: If the logarithmic representation LogðUðt, sÞ + κIÞ is true, is the linearized equation, where note that aðt, sÞ includes a parameter κ ∈ ℂ. If the logarithmic representation is true for κ = 0, the infinitesimal generator of linearized problem is simply represented by ∂ t aðt, sÞj κ=0 ∂ t LogðUðt, sÞÞ.
The condition for obtaining the linearized evolution equation is the locality for the evolution direction t, which leads to the boundedness of the spectral set of Uðt, sÞ. The theoretical procedure of obtaining the linearized problem is summarized as follows. For nonzero κ ∈ ℂ, first, Uðt, sÞ is regarded as an exponential function; second, calculating the logarithm of Uðt, sÞ; and finally the linearized operator generates the regularized evolution operator It is more clearly understood by the case of κ = 0, Consequently, the operator-logarithm is regarded as a mapping from "continuous group" to "bounded algebra." These alternative equations can be used to analyze quasilinear evolution equations and full-nonlinear evolution equations [28].
3.3.1. Autonomous Case. The regularity results have not been much studied in the Cauchy problem of hyperbolic partial differential equations (for a textbook, see [29]). The regularized evolution operator, which is also applicable to some hyperbolic type equations, is utilized to solve autonomous Cauchy problems.
in X, where AðtÞ ∈ GðXÞ: Y ⟶ X is assumed to be an infinitesimal generator of Uðt, sÞ satisfying the semigroup prop-erty, −T ≤ t, s ≤ T is satisfied, Y is a dense subspace of X permitting the representation shown in equation (19), and u s is an element of X.
As seen in equation (57), under the assumption of commutation, a related Cauchy problem is obtained as in X, where ∂ t aðt, sÞ = ∂ t LogðUðt, sÞ + κIÞ is well-defined. It is possible to solve the rewritten Cauchy problem, and the solution is represented by for u s ∈ X (cf. equation (58)).
Theorem 12. For u s ∈ X, there exists a unique solution uð·Þ ∈ Cð½−T, T ; XÞ for (68) with a convergent power series 9 Advances in Mathematical Physics representation: where κ is a certain complex number.
Proof. The unique existence follows from the assumption for AðtÞ. The regularized evolution operator e aðt,sÞ is holomorphic function (Theorem 11) with the convergent power series representation (equation (58)). By applying Uðt, sÞ + κI = e aðt,sÞ , the solution of the original Cauchy problem is obtained as for the initial value u s ∈ X. Note that AðtÞ is not assumed to be a generator of analytic evolution family but only a generator of evolution family [Q.E.D.: Theorem 12]. ?
For I λ denoting the resolvent operator of AðtÞ, the evolution operator defined by the Yosida approximation is written by so that more informative representation is provided by Theorem 12 compared to the standard theory based on the Hille-Yosida theorem.

Nonautonomous
Case. Series representation in autonomous part leads to the enhancement of the solvability. Let Y be a dense subspace of X permitting the representation shown in equation (19), and u s is an element of X. The regularized evolution operator is utilized to solved nonautonomous Cauchy problems.
in X, where AðtÞ ∈ GðXÞ: Y ⟶ X is assumed to be an infinitesimal generator of Uðt, sÞ satisfying the semigroup property and f ∈ L 1 ð−T, T ; XÞ is locally H€ older continuous on for a certain positive constant C H , γ ≤ 1 and −T ≤ t, s ≤ T.
The solution of nonautonomous problem does not necessarily exist in such a setting (in general, f ∈ Cð½−T, T ; XÞ is necessary).
Theorem 13. Let f ∈ L 1 ð−T, T ; XÞ be locally H€ older continuous on ½−T, T. For u s ∈ X, there exists a unique solution uð· Þ ∈ Cð½−T, T ; XÞ for (78) such that using a certain complex number κ.
Proof. Let us begin with cases with f ∈ Cð½−T, T ; XÞ. The unique existence follows from the standard theory of evolution equation. The representation follows from that of Uðt, sÞ and the Duhamel's principle where the convergent power series representation of e aðt,sÞ is valid (cf. equation (58)).
Next, let us consider cases with the locally H€ older continuous f ðtÞ. According to the linearity of equation (78), it is sufficient to consider the inhomogeneous term. For ϵ satisfy- is true by taking ϵ ⟶ 0. On the other hand, Advances in Mathematical Physics where ∂ τ Uðt, τÞ = −AðτÞUðt, τÞ is utilized. The last identity is obtained by applying AðtÞ = ðI + κUðs, tÞÞ∂ t aðt, sÞ. The H€ older continuity and equation (73) lead to the strong convergence of the right hand of equation (84): (due to ϵ ⟶ 0) for f ∈ L 1 ð0, T ; XÞ. AðtÞ is assumed to be an infinitesimal generator so that AðtÞ is a closed operator from Y to X. It follows that The right hand side of this equation is strongly continuous on ½−T, T. Consequently, It is seen that Ð t s ½e aðt,τÞ − κIf ðτÞdτ satisfies equation (78) and that it is sufficient to assume f ∈ L 1 ð0, T ; XÞ as H€ older continuous [Q.E.D.: Theorem 13].
? More simply, the unique solvability of nonautonomous case can be regarded in the context of decomposing the mild solution (for this terminology, see [18]).

Corollary 14.
Let f ∈ L 1 ð−T, T ; XÞ be locally H€ older continuous on ½−T, T. For u s ∈ X, there exists a unique solution uð⋅ Þ ∈ Cð½−T, T ; XÞ for (78) such that using a certain complex number κ.
Proof. The representation is regarded as The former part in the parenthesis is the mild solution of ∂ t uðtÞ = aðt, sÞuðtÞ + f ðtÞ, and the latter part in another parenthesis is the mild solution of ∂ t uðtÞ = f ðtÞ. The unique existence of mild solution for the former part is valid for H€ o lder continuous f ∈ L 1 ð0, T ; XÞ and that for the latter part is valid for any f ∈ L 1 ð−T, T ; XÞ [Q.E.D.: Corollary 14].
? Corollary 14 shows the meaning of introducing the alternative infinitesimal generator. This result should be compared to the standard theory of evolution equations in which the inhomogeneous term f is assumed to be continuous on ½−T, T. Consequently, in a purely abstract framework, the maximal regularity effect [31,32] is found in the solutions of renormalized evolution equations. In this sense, the alternative infinitesimal generator brings about the analytic semigroup theory for nonparabolic evolution equations.

Relativistic Formulation of Abstract
Evolution Equations 4.1. Formalism. The relativistic formulation of abstract evolution equations (55) is introduced to establish an abstract version of the Cole-Hopf transform in Banach spaces and to explain the nonlinear relation between the evolution operator and its infinitesimal generator [11]. The relativistic formulation is introduced for changing the evolution direction, which is necessary to justify the generalized Cole-Hopf transform. In this paper, the logarithmic representation of infinitesimal generator is utilized to formulate the relativistic form of abstract evolution equations. Here, the terminology "relativistic" is used in the sense that there is no especially dominant direction. In particular, the role of t direction (time direction) is not the absolute direction being compared to the other directions: x, y, and z directions (spatial directions) in the standard notation. While the relativistic treatment is associated with the equally valid time-reversal and spatial-reversal symmetries, here the relativistic form to the generalized 11 Advances in Mathematical Physics framework (57) is introduced without assuming the invertible property of evolution operators.
Let the standard space-time variables ðt, x, y, zÞ be denoted by ðx 0 , x 1 , x 2 , x 3 Þ receptively. It is further possible to generalize space-time variables to ðx 0 , x 1 , x 2 , x 3 , ⋯, x n Þ being valid to general ðn + 1Þ-dimensional space-time. In spite of the standard treatment of abstract evolution equations, the direction of evolution does not necessarily mean time-variable t = x 0 in the relativistic formulation of the abstract evolution equations. Consequently, the equal treatment of any direction and the introduction of multidimension are naturally realized by the relativistic formulation.

Definition 15 (relativistic form). For an evolution family of operators
where Y i is defence subspace of X i . The relativistic form of abstract evolution equations is defined as in X i , where X i is a functional space consisting of functions with variables x j with 0 ≤ j ≤ n skipping only j = i. Consequently, the unknown function is represented by For k ≠ i, let us begin with the abstract Cauchy problem  (90) and (91), the evolution direction is not limited to x 0 . This gives a reason why the formulation shown in equation (90) is called the relativistic form of abstract evolution equations. It means that if invertible evolution operator is obtained for one direction, the evolution operator for the other direction is not necessarily be the invertible. Here is a reason why it is useful to introduce a relativistic form based on the generalized logarithmic representation (cf. Corollary 8).
One utility of considering the evolution towards spatial direction is to explain and generalize the Cole-Hopf transform. For this purpose, it is necessary to realize the logarithmic representation of the infinitesimal generators defined in the relativistic form of the abstract evolution equations. That is, for a significant introduction of the relativistic form, it should be introduced together with the logarithmic representation. The condition to obtain the logarithmic representation is stated as follows.

Theorem 16. (relativistic form of logarithmic representation).
Let i denote any direction satisfying 0 ≤ i ≤ n. Let x i and ξ i satisfy −L ≤ x i , ξ i ≤ L and Y i be a dense subspace of a Banach space X i . A two-parameter evolution family of operators fUðx i , ξ i Þg −L≤x i ,ξ i ≤L satisfying equation (7) is assumed to exist in a Banach space X i (i.e., the inverse of Uðx i , ξ i Þ is not assumed). Under the existence of the preinfinitesimal generator Kðx i Þ: Y i ⟶ X i of Uðx i , ξ i Þ for the x i direction, let Uðx i , ξ i Þ and Kðx i Þ commute. The logarithmic representation of infinitesimal generator is obtained; there exists a certain complex number κ ≠ 0 such that where u ξ is an element in Y i , κ is taken from the resolvent set of Uðx i , ξ i Þ, and aðx i , ξ i Þ = LogðUðx i , ξ i Þ + κIÞ. Note that Uð x i , ξ i Þ is not assumed to be invertible.
Proof. Different from the proof of Theorem 3, here the similar statement is proved without assuming the invertible prop- The key point is that ðI − κe −aðx i ,ξ i Þ Þ −1 exists for a certain κ ∈ ℂ, even if Uðx i , ξ i Þ −1 does not exist. In particular, the obtained representation is more generally compared to the one obtained in Ref. [8]. For any Uðx i , ξ i Þ, operators LogðUðx i , ξ i Þ + κIÞ and LogðUðx i + h, ξ i Þ + κIÞ are well defined for a certain κ. The x i -differential in a weak sense is 12 Advances in Mathematical Physics formally written by where Γ, which is taken independent of x i , ξ i , and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Uðt, sÞ + κI and Uðt + h, sÞ + κI.
The discussion, which is the same as that shown in Theorem 3, leads to for u ∈ Y i . Because it is allowed to interchange Kðx i Þ with Uðx i , ξ i Þ, due to the integration by parts, where the details of procedure is essentially the same as Ref. [8]. It leads to for u ξ i ∈ Y i . It is notable that ðUðx i , ξ i Þ + κIÞ −1 is always welldefined for any κ taken from the resolvent set of Uðx i , ξ i Þ, 16]. ?
Under the existence of logarithmic representation for K ðx i Þ, the related concepts such as Þũ are similarly well-defined in the relativistic framework

Generalization of the Cole-Hopf
Transform. Now it is ready for establishing the general version of the Cole-Hopf transform. It corresponds to an application example of relativistic formulation provided. The Cole-Hopf transform [33][34][35][36][37] is a concept bridging the linearity and the nonlinearity. In the following, such a linear-nonlinear conversion relation is found within the relation between the infinitesimal generators and the generated semigroups. For t ∈ ℝ + and x ∈ ℝ, the Cole-Hopf transform reads where uðt, xÞ denotes the solution of linear equation and ψð t, xÞ is the solution of transformed nonlinear equation. On the other hand, e.g., for t ∈ ℝ and x ∈ ℝ n , the logarithmic representation of infinitesimal generator has been obtained in the abstract framework, where Uðx i , ξ i Þ denotes the evolution operator and Kðx i Þ is its infinitesimal generator. By taking a specific case with κ = 0, the similarity between them is clear. That is, the process of obtaining infinitesimal generators from evolution operators is expected to be related to the emergence of nonlinearity. Based on the logarithmic representation of infinitesimal generators obtained in the Banach spaces, the Cole-Hopf transform is generalized in the following sense: where in order to realize these features, the relativistic formulation of abstract evolution equation is newly introduced. Since the logarithmic representation shows a relation between an evolution operator and its infinitesimal generator, the correspondence to the Cole-Hopf transform means a possible appearance of nonlinearity in the process of defining an infinitesimal generator from the evolution operator. The next theorem follows.

Theorem 17. (generalization of the Cole-Hopf transform). Let
i be an integer satisfying 0 ≤ i ≤ n and Y be a dense subspace of a Banach space X. Let an invertible evolution family a Banach space X. Uðx i , ξ i Þ and Aðx i Þ are assumed to commute. For any u ξ i ∈ Y ⊂ X, the logarithmic representation is the generalization of the Cole-Hopf transform, where the logarithmic representation is obtained in a general Banach space framework, κ ≠ 0 is a complex number, and where aðx i , exists for a given interval 0 ≤ x i , ξ i ≤ L, its normalization defined in X corresponds to ψðt, xÞ.
Proof. The proof consists of five steps.
Step 1. Formulation. It is necessary to recognize the evolution direction of the heat equation as x, because the derivative on the spatial direction x is considered in the Cole-Hopf transform. The Cole-Hopf transform acts on one-dimensional heat equation where UðtÞ is a semigroup generated by μ −1/2 ∂ 2 x under the Dirichlet-zero boundary condition.
By changing the evolution direction from t to x, the heat equation is considered for x direction, where v 0 ðtÞ and v 1 ðtÞ are given initial functions. To establish the existence of semigroup for the x direction, it is sufficient to consider the generation of semigroup in L 2 ð−∞, ∞Þ by generalizing t -interval from ð0, ∞Þ to ð−∞, ∞Þ. The Fourier transform leads to where ω is a real number. Indeed, the following transforms are implemented. By solving the characteristic equation Advances in Mathematical Physics Meanwhile, based on the relativistic treatment, onedimensional heat equation is written as Let a linear operator A be defined by in L 2 ð−∞, ∞Þ × L 2 ð−∞, ∞Þ and the domain space of A be where H 1 ð−∞, ∞Þ = fuðtÞ∈L 2 ð−∞, ∞Þ ; u ′ ðtÞ∈L 2 ð−∞, ∞Þ , uð0Þ = 0g is a Sobolev space. The Fourier transform means that the diagonalization of A is equal tõ In this context, the master equation of the problem (105) is reduced to the abstract evolution equation in L 2 ð−∞, ∞Þ × L 2 ð−∞, ∞Þ, wherẽ It suggests that the evolution operator of equation (105) is generated by so that it is sufficient to show ±ðμ 1/2 ∂ t Þ 1/2 , as the infinitesimal generator. Note that the operatorÃ is not necessarily a generator of analytic semigroup, because the propagation of singularity should be different if the evolution direction is different. Consequently, the existence of semigroup for (116) in the x direction is reduced to show ±ðμ 1/2 ∂ t Þ 1/2 , as the infinitesimal generator in L 2 ð−∞, ∞Þ. In the following, the property of μ 1/2 ∂ t is discussed in the second step, and the fractional power of ðμ 1/2 ∂ t Þ 1/2 is studied in the third step.
Step 2. First order differential operator. The following lemma is proved in this step.

Proof of Lemma 18 Let λ be a complex number satisfying
in one-dimensional interval ð−∞, ∞Þ is a first-order ordinary differential equation with a constant coefficient, and the global-in-t solution necessarily exists for a given uð0Þ = u 0 ∈ ℂ. That is, λ/μ 1/2 is included in the resolvent set of ∂ t for an arbitrary complex number λ so that ðλ − μ 1/2 ∂ t Þ −1 is concluded to be well-defined in H 1 ð−∞, ∞Þ. Second, the resolvent operator ðλ − μ 1/2 ∂ t Þ −1 is estimated from the above. Since λ/μ 1/2 is included in the resolvent set of ∂ t , it is readily seen that ðλ − μ 1/2 ∂ t Þ −1 is a bounded operator on L 2 ð−∞, ∞Þ. More precisely, let us consider equation (119) being equivalent to If the inhomogeneous term satisfies f ðtÞ ∈ L 2 ð−∞, ∞Þ, 15 Advances in Mathematical Physics satisfies equation (120). According to the Schwarz inequality, is obtained, and the equality is positive if Reλ > 0 is satisfied. Its application leads to Further application of the equality results in for Reλ > 0, and therefore, follows. That is, for Reλ > 0, is valid. The surjective property of λI − μ 1/2 ∂ t is seen by the unique existence of solutions u ∈ L 2 ð−∞, ∞Þ for the Cauchy problem of equation (120). A semigroup is generated by taking a subset of the complex plane as where Ω is included in the resolvent set of μ 1/2 ∂ t . For λ ∈ Ω , ðλI − μ 1/2 ∂ t Þ −1 exists, and is obtained. Consequently, according to the Lumer-Phillips theorem [38] for the generation of quasi contraction semigroup, μ 1/2 ∂ t is confirmed to be an infinitesimal generator in L 2 ð−∞, ∞Þ, and the unique existence of global-in-x weak solution follows [Q.E.D.: Lemma 18]. ? The semigroup generated by μ 1/2 ∂ t is represented by so that the group is actually generated by μ 1/2 ∂ t . Indeed, the similar estimate as equation (130) can be obtained for Reλ that should be compared to equation (121).
Step 3. Fractional powers of operator. The following lemma is proved in this step.
and hence as using the commutation assumption. The nonlinear Anzatz −2μ −1/2 ð∂ x uðt, xÞÞuðt, xÞ −1 of the Burgers' equation is essentially represented by in the abstract form. The similarity between equation (101) and the standard definition of operator norm is clear. In particular, the evolution direction is generalized from x to x i in equation (101).
Step 5. The generalized Cole-Hopf transform (100) shows that the nonlinearity of semigroup can appear simply by altering the evolution direction under a suitable identification between the infinitesimal generator and the evolution operator. In this sense, equation (90) is regarded as a local-in-x i linearized equation, if Uðx i , ξ i Þ is a nonlinear semigroup (semigroup related to the nonlinear equations). Furthermore, the generalized Cole-Hopf transform (100) suggests that the relation between evolution operator and its infinitesimal generator corresponds essentially to the transform between linearity and nonlinearity. In the same context of generalizing Miura transform between the Korteweg-de-Vries and the modified Korteweg-de-Vries equations, the logarithmic representation is utilized [15].

B ðXÞ-Module.
The algebraic structure is studied based on the relativistic form of abstract equations. The operator aðx i , ξ i Þ = LogðUðx i , ξ i Þ + κIÞ is bounded on X. It follows that e aðx i ,ξ i Þ is well-defined by the convergent power series. Note again that e aðx i ,ξ i Þ can be defined without assuming welldefined ∂ t a ðx i ,ξ i Þ . Even without taking into account the detail property of the infinitesimal generator ∂ t a ðx i ,ξ i Þ , the exponentiability is realized by the boundedness of aðx i , ξ i Þ. In this section, beginning with aðx i , ξ i Þ = LogðUðx i , ξ i Þ + κIÞ, an algebraic module over a Banach algebra is defined. The essential idea of presenting a useful algebraic structure is not to examine directly the set of ∂ t aðx i , ξ i Þ but to focus on the set of aðx i , ξ i Þ at first, and then, the algebraic structure of infinitesimal generators is discovered in the next. Although aðx i , ξ i Þ ∈ BðXÞ is trivially the infinitesimal generator, what is explained here is the structure of the set of preinfinitesimal generators ∂ t aðx i , ξ i Þ.
Theorem 20. (normed vector space). Let U j ðx i , ξ i Þ be evolution operators satisfying equation (19) and Log U j ðx i , ξ i Þ be well-defined for any x i , ξ i ∈ ½−L, L and j = 1, 2, ⋯, n. Log U j ðx i , ξ i Þ is assumed to commute with each other.
is a normed vector space over the complex number field, where BðXÞ denotes a set of all the bounded operators on X.
Proof. In case of κ = 0, the operator aðx i , ξ i Þ is reduced to 17

Advances in Mathematical Physics
The operator sum is calculated using the Dunford-Riesz integral then, the sum closedness is clear. Here, Γ′ is assumed to be included in Γ, and this condition is not so restrictive in the present setting. In a different situation, when Uðt, rÞ and V ðt, rÞ commute for the same t and r, another kind of sum is calculated as where for and then, the sum closedness is clear. Although the logarithm function is inherently a multivalued function, the uniqueness of sum operation is ensured by the single-valued property of the principal branch "Log." Consequently, since the closedness for scalar product is obvious, wherẽ holds the semigroup property in a similar way to Wðx i , η i Þ, Consequently, V Lg ðXÞ is a normed vector space over the complex number field. In particular, the zero operator LogI is included in V Lg ðXÞ. Theorem 1 has been proved [Q.E.D.: Theorem 20]. ?
Theorem 21. (BðXÞ-module). Let U j ðx i , ξ i Þ be evolution operators satisfying equation (19) for any x i , ξ i ∈ ½−L, L and j = 1 , 2, ⋯, n. For a certain K ∈ BðXÞ, let a subset of BðXÞ in which each element is assumed to commute with LogðU j ðx i , ξ i Þ + KÞ are assumed to commute with each other.
is a module over the Banach algebra.
Proof. It is worth generalizing the above normed vector space. In this sense, utilizing a common operator K ∈ BðXÞ, components are changed to LogðUðx i , ξ i Þ + KÞ: The operator sum is calculated as

Advances in Mathematical Physics
After introducing a certain K ∈ BðXÞ with sufficient large ∥K∥, it is always possible to take integral path Γ ′ to be included in Γ. Since the part "KUðt, rÞ + KUðr, sÞ + K 2 I" is included in BðXÞ, the sum-closedness is clear. In a different situation, when U 1 ðt, rÞ and U 2 ðt, rÞ commute for the same t and r, another kind of sum is calculated as Since the part " is included in BðXÞ, the sum-closedness is clear.
The product KLogðU 1 ðx i , ξ i Þ + K 2 Þ ∈ BðXÞ is justified by the operator product equipped with BðXÞ. Since the closedness for operator product within BðXÞ is obvious, wherê holds the semigroup property, and using an identity operator I ∈ BðXÞ, and therefore are valid Consequently, B Lg ðXÞ is a module over a Banach algebra. In particular, a relation V Lg ðXÞ ⊂ B Lg ðXÞ is satisfied. The statement has been proved [Q.E.D.: Theorem 21]. ?
The next corollary follows.

Corollary 22.
(BðXÞ-module for infinitesimal generators). Let U j ðx i , ξ i Þ be evolution operators satisfying equation (19) for any x i , ξ i ∈ ½−L, L and j = 1, 2, ⋯, n. For a certain K ∈ BðXÞ, let a subset of BðXÞ in which each element is assumed to commute with LogðU j ðx i , ξ i Þ + KÞ be B ab ðXÞ. LogðU j ðx i , ξ i Þ + KÞ are assumed to commute with each other.
is a module over a Banach algebra.
Proof. According to the linearity of differential operator, the introduction of differential operator ∂ x i is true without any additional treatment. It is sufficient to see that there exists a certain η i such that according to the mean value theorem ðξ i ≤ η i ≤ x i Þ [Q.E.D.: Corollary 22]. ?
The module over a Banach algebra is called BðXÞ-module. For the structure of B Lg ðXÞ, a certain originally unbounded part can be classified to LogðUðx i , ξ i Þ + KÞ ∈ Bð XÞ and the rest part to K ∈ B ab ðXÞ. Here, the terminology "originally unbounded" is used, because some unbounded operators are reduced to bounded operators under the validity of the logarithmic representation.
It is necessary to connect the concept of BðXÞ-module to the set of infinitesimal generators. Let us move on to the operator ∂ x i ½KLogðU j ðx i , ξ i Þ + KÞ, which is expected to be the preinfinitesimal generator of exp ½KLogðU j ððx i , ξ i ÞÞ + KÞ: This property is surely true by the inclusion relation B Lg ðXÞ ⊂ BðXÞ. It is also suggested by the inclusion relation B Lg ðXÞ ⊂ BðXÞ, operators ∂ x i LogðU j ððx i , ξ i ÞÞ + KÞ are the preinfinitesimal generators if LogðU j ððx i , ξ i ÞÞ + KÞ ∈ G Lg ðX Þ is satisfied. Consequently, the unbounded sumperturbation for infinitesimal generators is seen by the sum closedness of BðXÞ-module. Note that it does not require the self-adjointness of the operator.

Advances in Mathematical Physics
The preinfinitesimal generator property is examined for products of operators in the next two theorems. Theorem 23. (product perturbation for preinfinitesimal generators). For a certain K ∈ BðXÞ, let a subset of BðXÞ in which each element is assumed to commute with LogðU i ðx i , ξ i Þ + KÞ be B ab ðXÞ. Let an operator denoted by be included in B ab ðXÞ, where the evolution operator Uðt, sÞ on X is generated by AðtÞ, L is an element in B ab ðXÞ, and K is an element in BðXÞ. Let K and L be further assumed to be independent of x i . The product of preinfinitesimal generators, which is represented by is also the preinfinitesimal generators in X, where aðx i , ξ i Þ = LogðUðx i , ξ i Þ + KÞ: is true. The basic calculi using the t-independence of K leads to the product of operator LAðtÞ. It is welldefined by under the commutation assumptions, where the relation ∂ x i Uðx i , ξ i Þ = Aðx i ÞUðx i , ξ i Þ is applied. Let x i , ξ i ∈ ½−L, +L satisfy ξ i < x i . The preinfinitesimal generator property of LAðtÞ is confirmed by where a certain real number σ ∈ ½ξ i , x i is determined by the mean value theorem. Consequently, due to the boundedness KÞ is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 23]. ?
The operator L can be regarded as a perturbation to the operators in B Lg ðXÞ. This lemma shows the product perturbation for the infinitesimal generators of C 0 -semigroups under the commutation, although the perturbation has been studied mainly for the sum of operators. It is remarkable that the self-adjointness of the operator is not required for this lemma. For the details of conventional bounded sum perturbation and the perturbation theory for the self-adjoint operators, see Ref. [42].
Theorem 24. (operator product). For a certain Kðx i Þ ∈ BðXÞ, let a subset of BðXÞ in which each element is assumed to commute with LogðUðx i , ξ i Þ + Kðx i ÞÞ be B ab ðXÞ. Let an operator denoted by be included in B Lg ðXÞ, where the evolution operator Uðx i , ξ i Þ on X is generated by Aðx i Þ, L is an element in B ab ðXÞ, and Kðx i Þ is an element in BðXÞ. Let L and Kðx i Þ be x i -independent and x i -dependent, respectively. The operators represented by is the preinfinitesimal generators in X, if the operator Lðη i Þ ∈ BðXÞ is strongly continuous with respect to η i in the interval ½ξ i , x i .

Advances in Mathematical Physics
Proof. Let x i , ξ i ∈ ½−L, +L satisfy ξ i < x i . The preinfinitesimal generator property is reduced to the possibility of applying the mean value theorem.
where a certain real number σ i ∈ ½ξ i , x i is determined by the mean value theorem. Consequently, Lðx i Þ∂ x i ½LLogðUðx i , ξ i Þ + Kðx i ÞÞ is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 24]. ?
Equation (160) provides one standard form for the representation of operator products in the sense of logarithmic representation. Consequently, BðXÞ-module is associated with the preinfinitesimal generator.

5.2.
Formulation of Rotation Group. The application example of BðXÞ-module is provided. The concept of BðXÞ-module is generally enough to provide a foundation of the conventional bounded formulation of Lie algebras (for a textbook, see [43]). In other words, by means of BðXÞ-module, the intersection of the Banach algebra (including only bounded operators) and the extracted bounded part of the Lie algebra (generally including unbounded operators) is shown. More precisely, using BðXÞ-module, the bounded part is extracted from unbounded angular momentum operators. The extracted bounded parts are utilized to formulate the rotation group with incorporating the unboundedness of angular momentum algebra.
The mathematical foundation of rotation group is demonstrated [14]. Although the evolution parameter in this paper is denoted by t, s ∈ ½−T, +T, it is more likely to be denoted by θ, σ ∈ ½−Θ, +Θ, because the evolution parameter in the present case means the rotation angle. The rotation group is generated by the angular momentum operator (for textbooks, see Refs. [44,45]). The angular momentum operator includes a differential operator, as represented by where ℏ is a real constant called the Dirac constant. The appearance of differential operator ∇ in the representation of L is essential. The operator ∇ is an unbounded operator for example in a Hilbert space L 2 ðR 3 Þ, while it must be treated as a bounded operator in terms of establishing an algebraic ring structure. Furthermore, the operator boundedness is also indispensable for some important formulae such as the Baker-Campbell-Hausdorff formula and the Zassenhaus formula to be valid. In general, the exponential of unbounded operators cannot be represented by the power series expansion (cf. the Yosida approximation in a typical proof of the Hille-Yosida theorem; e.g., see Ref. [22]). Let ℝ 3 be the three-dimensional spatial coordinate spanned by the standard orthogonal axes, x, y, and z. The angular momentum operator L is considered in L 2 ðℝ 3 Þ. The angular momentum operator consists of x, y, and z components L x = ih y∂ z − z∂ y À Á , respectively. The commutation relations are true, where ½L i , L j ≔ L i L j − L j L i denotes a commutator product ði, j = x, y, zÞ. The commutation of angular momentum operators arises from the commutation relations of the canonical quantization Indeed, the momentum operator p = ðp x , p y , p z Þ is represented by p = ihð∂ x , ∂ y , ∂ z Þ in quantum mechanics. It is remarkable that the commutation is always true for the Newtonian mechanics; i.e., ½x, p x = ½y, p y = ½z, p z = 0 is true in addition to ½y, p x = ½y, p z = ½z, p x = ½z, p y = ½x, p y = ½x, p z = 0: Let a set of all bounded operators on L 2 ðℝ 3 Þ be denoted by BðL 2 ðℝ 3 ÞÞ. A set of operators fL k ; k = x, y, zg or fiL k / h ; k = x, y, zg with the commutation relation (165) is regarded as the Lie algebra. In particular fb αL x + b βL y + b γ L z ; b α, b β, b γ ∈ ℂg forms a vector space over the complex number field, while fαðiL x /hÞ + βðiL y /hÞ + γðiL z /hÞ ; α, β , γ ∈ ℝg is a vector space over the real number field. It is possible to associate the real numbers α, β, and γ with the Euler angles (for example, see Ref. [46]). The second term of the right hand side of disappears as far as the commutator product ½L i , L j is