Fractional Powers of Infinitesimal Generators of Semigroups

Foundations of Fractional Dynamics: A Short Account

“…3.6] Frequently an integral representation §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].…”

“…Despite the long history of fractional calculus in mathematics (see [2][3][4][5][6] for reviews), despite numerous publications on fractional powers of infinitesimal generators [7][8][9][10][11][12][13][14][15][16], and despite a rapidly growing literature on possible applications of fractional dynamical systems to physical phenomena (see [17][18][19][20] and the present volume for reviews), there seem to exist only few publications discussing the physical foundations of fractional dynamics. a [207.…”

Threefold Introduction to Fractional Derivatives

“…for every f ∈ B for which the limit exists in the norm of B [93,120,121,123]. This approach is clearly inspired by the Marchaud form (2.82).…”

Fractional powers of operators, K-functionals, Ulyanov inequalities

“…Functional calculus. To explain fractional powers of operators associated with the infinitesimal generator of an equibounded (C 0 )-semigroup we use a functional calculus due to Laurent Schwartz [6] (see also [9]) which is an elegant and powerful tool for our purposes. Concerning an approach to fractional powers of operators in the framework of the more general class of non-negative operators we refer, e.g., to the book of Martinez and Sanz [4].…”