### N-th roots of a non-negative operator: conditions for uniqueness.

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If -A is the generator of an equibounded ${C}_{0}$-semigroup and 0 < Re α < m (m integer), its fractional power ${A}^{\alpha}$ can be described in terms of the semigroup, through a formula that is only valid if a certain function ${K}_{\alpha ,m}$ is nonzero. This paper is devoted to the study of the zeros of ${K}_{\alpha ,m}$.

For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, $({(-\Delta )}^{\alpha}u,\varphi )=(u,{(-\Delta )}^{\alpha}\varphi )$, α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean...

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the ${L}^{p}\left(\mathbb{R}\right)$-spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative...

Mathematics Subject Classification: Primary 47A60, 47D06. 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) + λm) 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...

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