Characterizations of K-Functionals Built from Fractional Powers of Infinitesimal Generators of Semigroups

Abstract

Abstract. On a Banach space X consider an equibounded (C_0)-semigroup of linear operators { T(t): t ≥ 0} with infinitesimal generator A . We introduce fractional powers (-A)^α, α >0 , of A with domain D((-A)^α )) and characterize the K -functionals with respect to (X,D((-A)^α)) via fractional differences [I-T(t)]^α , via appropriate truncated hypersingular integrals and via some type of fractional integral over the resolvent of A . Immediate consequences are an abstract Marchaud-type inequality for moduli of smoothness arising from (semi-) groups of operators as well as optimal and nonoptimal approximation results.