The Cholesky algorithm allows one to factor positive semidefinite matrices or positive operators on the space of square summable sequences as LL* with L lower triangular. The story is very different if one wants instead to do a UU* factorisation with U upper triangular. But this UU* factorisation turns out to be ``natural" for Toeplitz operators on the Hardy space.

Necessary and sufficient conditions for when one can do UU* factorisations come from the theory of RKHS and easily extend to the multivariable case. In this talk we will present this theory, some recent applications, and discuss open problems.