Fractional powers of generators of equicontinuous semigroups and fractional derivatives

Abstract

We analyze fractional powers H", a > 0, of the generators H of uniformly bounded locally equicontinuous semigroups 5.

Representation of complex powers of C-sectorial operators

“…also [35,Subsect. 3.2] and [31]). We also want to mention the works of W. Lamb [29,30] on fractional powers of operators on Frechét spaces following that of A.C. McBride [36].…”

Fractional Powers of Infinitesimal Generators of Semigroups

“…This application of Schwartz' functional calculus is mentioned, for instance, by Faraut [9]. In the late eighties, the subject was taken up by Lanford and Robinson [24]. They analyzed fractional powers of infinitesimal generators starting from the definition according to the functional calculus due to Schwartz.…”

“…In [46] this result is deduced from an asymptotic formula for power functions of fractional order due to Ingham [17]. For an alternative proof using Fourier transform methods we refer to [24] and [37].…”

Fractional powers of momentum of a spectral distribution

“…Take A the generator of a uniformly bounded semigroup {G(t)}t>o ; then in lieu of constructing Aa as the generator of the semigroup {Ga(t)}r>o one can show that Aa could be the ath derivative of A , i.e., Aax = lim((G(t)-I)/t)ax, where this limit exists if and only if x e D(Aa) (see, for example, [17]). …”

“…In §4, using the relation of the momentum of a spectral distribution with regularized group, we give the fractional derivatives formula of (iB)a (see [17]). …”