Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator $A$ and its adjoint $A^*$? I am interested in finding such a decomposition, but I could not get one by simple algebraic manipulations. Does anybody know if something like that exists?

So again, I want to get: $- \Delta = A^* A$ on the $\mathbb{S}^2$. In case that you know a way to get this, I would be also interested in possible domains(if this is non-trivial), such that $A$ is closed.

I think, in principle, the chances that something like that exists are fairly good, as the Laplacian is positive and self-adjoint.

Edit: I know that there is a (i.e. unique positive) square root for these kinds of operators, but I am rather looking for an analogue to the decomposition $-\frac{d^2}{dx^2}= (- \frac{d}{dx}) ( \frac{d}{dx}).$ So, I am rather looking for a casual root( in the sense that the root is comfortable for calculations). I hope I could give you an idea, what I am looking for.