Representation of fractional powers of infinitesimal generators of semigroups

Representation of complex powers of C-sectorial operators

“…Albeit it seems very plausible to be true, it is not clear whether a modification of the above result continues to hold if γ ∈ C + \ (0, ∞) and γ < r, since it is not clear whether in this case we have that ∞ 0 |q γ,r (u)| du < ∞, with q γ,r (u) being the function defined in the fundamental lemma of [7]. Theorem 2.2.…”

“…Therefore, the assertion (i) is an immediate consequence of Proposition 2.1. we would like to mention the papers [7] by H. Berens, P.L. Butzer, U. Westphal and [39] by S. Samko.…”

“…Butzer, U. Westphal and [39] by S. Samko. Making use of the method proposed in [7], one can prove the following: Suppose 0 < γ < r, where r ∈ N, and −A is the integral generator of an equicontinuous C-regularized semigroup (T (t)) t≥0 . Then, for every x ∈ D(A γ ), we have that the limit y = lim…”

“…Balakrishnan [5]- [6], H. Berens, P.L. Butzer, U. Westphal [7], H. Komatsu [23], S. Samko [39] and some other authors mentioned later (cf. also the monograph [35] by C. Martínez -M. Sanz [35] for comprehensive exposition of results), established for strongly continuous semigroups of operators, and H.O.…”

Fractional powers of generators of equicontinuous semigroups and fractional derivatives

“…A comprehensive description of the subject is also given in the series of papers by Komatsu [11]. Most of this work concerns generators of strongly continuous semigroups, or operators with strongly continuous resolvent families, but Berens, Butzer, Westphal [3] and Komatsu [11] have also derived results in the case of weak* continuity. Our methods have the advantage of unifying these two cases, and extending the results in several ways.…”