Boundary values of analytic semigroups generated by fractional Laplacians

In the present paper, using the theory of boundary values of analytic semigroups, we find necessary and sufficient conditions to guarantee that the operator \(i(-\Delta )^{{\alpha }/{2}}\) $i{(-\mathrm{\Delta })}^{\alpha /2}$ generates a strongly continuous semigroup in \(L^p(\mathbb {R}^n)\) $L^p({\mathbb{R}}^n)$ .