To the best of my knowledge the problem at hand is a generalisation of monic matrix polynomials. Can a closed form solution to the following equation be found,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + u_1A_1XB_1 + u_0C = 0,$$
where, $A_i, B_i$ and $C$, are known linear operators (on potentially infinite dimensional Hilbert spaces) and $u_i\in \mathit{\mathbb{R}}$ are finite scalars? How would one go about finding the roots (or solvents) of such a polynomial?
Perhaps, it is helpful to note that the solution for $X$ is expected to be a positive semi-definite self-adjoint trace class operator.