Our aim here is to collect and to compare two definitions of the fractional powers of non-negative operators that can be found in the literature; we will present the proof of an equivalence and compare properties of that notions in different approaches. Then we will apply next this equivalence in the study of global and local existence of solution for the semilinear Cauchy problem in $\R^N$ with fractional Laplacian \[ eft\{ \begin{array}{c} u_t = -(-\Delta)^lpha u + f(x,u), u(0,x) = u_0(x), \quad x ı \R^N. \end{array} \right. \]