*2.3. Logarithmic Representation of Infinitesimal Generators*

Let *X* be a Banach space, *B*(*X*) is a set of bounded linear operators on *X*, and *Y* be a dense Banach subspace of *X*. The Cauchy problem for the first order abstract evolution equation of hyperbolic type [9,10] is defined by

$$\begin{cases} du(t)/dt - A(t)u(t) = f(t), \qquad t \in [0, T] \\ u(0) = u\_0 \end{cases} \tag{6}$$

in *X*, where *A*(*t*) : *Y* → *X* is assumed to be the infinitesimal generator of the evolution operator *U*(*t*,*s*) ∈ *B*(*X*) satisfying the strong continuity and the semigroup property:

$$\mathcal{U}(t,s) = \mathcal{U}(t,r)\mathcal{U}(r,s)$$

for 0 ≤ *s* ≤ *r* ≤ *t* < *T*. *U*(*t*,*s*) is a two-parameter *C*0-semigroup of operator (for definition, see [11–13]) that is a generalization of one-parameter *C*0-semigroup and therefore an abstract generalization of the exponential function of operator. If *A*(*t*) is confirmed to be an infinitesimal generator, then the solution *u*(*t*) is represented by *u*(*t*) = *U*(*t*,*s*)*us* with *us* ∈ *X* for a certain 0 ≤ *s* ≤ *T* (cf. Hille-Yosida Theorem; for example, see [11–13]). Then, for a certain complex number *κ*, the alternative bounded infinitesimal generator *a*(*t*,*s*) = Log(*U*(*t*,*s*) + *κI*) to *A*(*t*) is well defined [14,15], where Log denotes the principal branch of logarithm.

**Lemma 1** (Logarithmic representation of infinitesimal generators [14])**.** *Let t and s satisfy* 0 ≤ *t*,*s* ≤ *T, and Y be a dense subspace of X. Let a*(*t*,*s*) ∈ *B*(*X*) *be defined by a*(*t*,*s*) = Log*U*(*t*,*s*)*. If A*(*t*) *and U*(*t*,*s*) *commute, infinitesimal generators* {*A*(*t*)}0≤*t*≤*<sup>T</sup> are represented by means of the logarithm function; there exists a certain complex number κ* = 0 *such that*

$$A(t)\,\,\mu = \left(I - \kappa e^{-a\langle t,\mathbf{s}\rangle}\right)^{-1} \partial\_{\mathbf{l}} a(t,\mathbf{s})\,\,\mu\,\,\tag{7}$$

*where u is an element of a dense subspace Y of X, and the logarithm of operator is defined by the Riesz-Dunford integral.*

**Proof.** Only formal discussion is given here (for the detail, see [14,16]). Since *a*(*t*) is defined by *a*(*t*,*s*) = Log(*U*(*t*,*s*) + *κI*), *∂ta*(*t*,*s*)=(*U*(*t*,*s*) + *κI*)−1*∂tU*(*t*,*s*),

$$
\partial\_t (\mathcal{U}(t, \mathbf{s}) + \kappa I) \partial\_l a(t, \mathbf{s}) = (\mathcal{U}(t, \mathbf{s}) + \kappa I)(\mathcal{U}(t, \mathbf{s}) + \kappa I)^{-1} \partial\_l \mathcal{U}(t, \mathbf{s}).
$$

Under the commutation relation between *U*(*t*,*s*) and *A*(*t*),

$$\begin{split} &A(t)\ u := \partial\_t \text{Log} \mathcal{U}(t, \mathbf{s}) u \\ &= \mathcal{U}(t, \mathbf{s})^{-1} \partial\_t \mathcal{U}(t, \mathbf{s}) u \\ &= \mathcal{U}(t, \mathbf{s})^{-1} (\mathcal{U}(t, \mathbf{s}) + \kappa I) \partial\_t a(t, \mathbf{s}) \ u \\ &= (I + \kappa (\boldsymbol{\epsilon}^{a(t, \mathbf{s})} - \kappa I)^{-1}) \partial\_t a(t, \mathbf{s}) \ u \\ &= (\boldsymbol{\epsilon}^{a(t, \mathbf{s})} - \kappa I + \kappa I) (\boldsymbol{\epsilon}^{a(t, \mathbf{s})} - \kappa I)^{-1} \partial\_t a(t, \mathbf{s}) \ u \\ &= (I - \kappa \boldsymbol{\epsilon}^{-a(t, \mathbf{s})})^{-1} \partial\_t a(t, \mathbf{s}) \ u , \end{split}$$

where *u* is an element in *Y*.

The commutation assumption is trivially satisfied if *A*(*t*) is independent of *t*. Equation (7) is the logarithmic representation of infinitesimal generator *A*(*t*). This representation is the generalization of the Cole-Hopf transform [4,5].
