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Zbl 1059.47501
Samko, Stefan G.
Approximative approach to fractional powers of operators. (English)
[A] Begehr, Heinrich G. W. (ed.) et al., Proceedings of the second ISAAC congress. Vol. 2. Proceedings of the International Society for Analysis, its Applications and Computation Congress, Fukuoka, Japan, August 16--21, 1999. Dordrecht: Kluwer Academic Publishers. Int. Soc. Anal. Appl. Comput. 8, 1163-1170 (2000). ISBN 0-7923-6598-4/hbk

Summary: A new formula is obtained for fractional powers $(-A)^\alpha$ of operators in a Banach space (which are generators of strongly continuous uniformly bounded semigroups $T_t$). This formula is based on the so-called approximative approach and represents the fractional power $(-A)^\alpha f$ as a limit of ``nice'' operators of the form $\int^\infty_0 u_\varepsilon(t) T_t f\,dt$ with the elementary function $u_\varepsilon(t)={d\over dt} [{t\over (t+ i\varepsilon)^{1+\alpha}}]$.

MSC 2000:: *47A60 Functional calculus of operators
47A58 Operator approximation theory
47D06 One-parameter semigroups and linear evolution equations
26A33 Fractional derivatives and integrals (real functions)

Keywords: fractional powers of operators; semigroups of operators; infinitesimal operator; Balakrishnan formula; fractional differentiation; Marchaud formula

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