Fractional powers of operators, $ K$-functionals, Ulyanov inequalities
Volume 88 / 2010
Banach Center Publications 88 (2010), 273-283
MSC: Primary 47D60, 46B70; Secondary 41A17, 41A65.
DOI: 10.4064/bc88-0-22
Abstract
Given an equibounded $\,{\rm (}{\cal C}_0{\rm )}$-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 $\,(X,D((-A)^\alpha)), \,\alpha >0,$ is characterized via the associated resolvent $\,R(\lambda;A).$ Under the assumption that the resolvent satisfies a Nikolskii type inequality, $\| \lambda R(\lambda ; A )f\| _Y \le c \varphi (1/\lambda ) \| f \|_X ,$ 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.
