[217.3.1] The operators GTαt form a family of strongly continuous
semigroups on C0G* provided that the
translations Tt inside the integral in eq. (28)
are strongly continuous [33, 48] and
∫0∞Tshαs/t/tds<∞.
[217.3.2] In this case the infinitesimal generators for
0<α≤1 are defined by

Aαϱ~=s-limt→0GTαtϱ~-ϱ~t | | (32) |

for all ϱ~∈C0G* for which the strong limit
s-lim exists.
[217.3.3] In general, the infinitesimal generators are unbounded operators.
[217.3.4] If A=-d/dt denotes the infinitesimal generator of
the translation Tt in eq. (28), then

are fractional time derivatives [50, 16].
[217.3.5] The action of Aα on mixed states can be represented
in different ways.
[217.3.6] Frequently an integral representation

Aαϱ~=limϵ→oC∫ϵ∞t-α-11-Ttϱ~dt | | (34) |

of Marchaud type [8, 51] is used.
[217.3.7] The integral representation

Aαϱ~=limϵ→oC∫ϵ∞t-αA1-tA-1ϱ~dt | | (35) |

[page 218, §0]
in terms of the resolvent of A [12] defines the same
fractional derivative operator [52].
[218.0.1] Representations of Grünwald-Letnikov type
are also well known [16].

[218.1.1] In summary, fractional dynamical systems must be expected to
appear generally in mathematical models of macroscopic phenomena.
[218.1.2] They arise as coarse grained
macroscopic time evolutions from inducing a microscopic time
evolution on the subsets G⊂Γ of small measure
in phase space, that are typically incurred
in statistical mechanics [1, 21, 50].