I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for $$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}.$$ My question is: why are fractional powers set to $-\Delta$ and not to $\Delta$? Is this somehow related to the spectrum of $-\Delta$ being positive?
Thank you in advance.