Joint Moments of Bilinear Resolvents

Given A,B∈ B(H)₊, we consider the existence and uniqueness of a positive operator C∈ B(H)₊ such that \[ (I+mA+nB+mnC)⁻¹=∫_{[0,1]²}sᵐtⁿdE(s,t), m,n≥0, \] 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.