Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter Trebels; Ursula Westphal

Banach Center Publications, Tome 89 (2010), p. 273-283 / Harvested from The Polish Digital Mathematics Library

Résumé

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, ${| | λ R (λ; A) f | |}_{Y} {\leq c φ (1 / λ) | | f | |}_{X}$ , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has good control on the resolvent but not on the semigroup.

Publié le : 2010-01-01

Zbl 1259.47056

EUDML-ID : urn:eudml:doc:282573

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     author = {Walter Trebels and Ursula Westphal},
     title = {Fractional powers of operators, K-functionals, Ulyanov inequalities},
     journal = {Banach Center Publications},
     volume = {89},
     year = {2010},
     pages = {273-283},
     zbl = {1259.47056},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc88-0-22}
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Walter Trebels; Ursula Westphal. Fractional powers of operators, K-functionals, Ulyanov inequalities. Banach Center Publications, Tome 89 (2010) pp. 273-283. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc88-0-22/