Fractional powers of operators, K-functionals, Ulyanov inequalities

Abstract

Abstract. Given an equibounded (C0)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to (X, D((−A) α )), α > 0, is characterized via the associated resolvent R(λ; A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, λR(λ; A)f Y ≤ cϕ(1/λ) f X , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has good control on the resolvent but not on the semigroup.1. Introduction. Let (X, · X ) be a Banach space, {T (t) : t ≥ 0} be an equibounded semigroup of class (C 0 ) of linear operators from X into itself, that is,