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Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter TrebelsUrsula Westphal — 2010

Banach Center Publications

Given an equibounded (₀)-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, | | λ R ( λ ; A ) f | | Y c φ ( 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...

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