Joint Moments of Bilinear Resolvents

Given $A,B\in B\left(H\right)_{+}$, we consider the existence and uniqueness of a positive operator $C\in B\left(H\right)_{+}$ such that \[ \left(I+mA+nB+mnC\right)^{-1}=\int_{\left[0,1\right]^{2}}s^{m}t^{n}dE\left(s,t\right),\quad m,n\geq0, \] for a positive operator-valued measure $E$. This is a joint Hausdorff moment problem. Without $C$, it holds if and only if $AB=BA$. In dimension two we give a complete criterion for noncommuting pairs. In higher finite dimensions, nonexistence is generic. Under certain spectral and cyclicity assumptions, we obtain a necessary and sufficient condition on arbitrary Hilbert spaces. The completion is unique in this case. We also classify these pairs up to unitary equivalence. The class includes irreducible infinite-dimensional examples, even with continuous spectrum.