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Approximative approach to fractional powers of operators. (English) Zbl 1059.47501
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 (ISBN 0-7923-6598-4/hbk). Int. Soc. Anal. Appl. Comput. 8, 1163-1170 (2000).
Summary: A new formula is obtained for fractional powers (-A) α 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) α f as a limit of “nice” operators of the form 0 u ε (t)T t fdt with the elementary function u ε (t)=d dt[t (t+iε) 1+α ].
MSC:
47A60Functional calculus of operators
47A58Operator approximation theory
47D06One-parameter semigroups and linear evolution equations
26A33Fractional derivatives and integrals (real functions)