Geodesic
Fractional Powers of Almost Non-Negative Operators
Fractional calculus and applied analysis, Tome 8 (2005) no. 2, pp. 201-230.

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In this paper, we extend the theory of complex powers of operators to a class of operators in Banach spaces whose spectrum lies in C ]−∞, 0[ and whose resolvent satisfies an estimate ||(λ + A)^(−1)|| ≤ (λ^(−1) + λ^m) M for all λ > 0 and for some constants M > 0 and m ∈ R. This class of operators strictly contains the class of the non negative operators and the one of operators with polynomially bounded resolvent. We also prove that this theory may be extended to sequentially complete locally convex spaces.
Keywords: Fractional Powers, Non-Negative Operators, Almost Sectorial Operators, Functional Calculus, Semigroups of Operators, 47A60, 47D06 
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Martínez, Celso; Sanz, Miguel; Redondo, Antonia. Fractional Powers of Almost Non-Negative Operators. Fractional calculus and applied analysis, Tome 8 (2005) no. 2, pp. 201-230. http://geodesic.mathdoc.fr/item/FCAA_2005_8_2_a6/