Computing extreme singular values of free operators

A recent development in random matrix theory, the intrinsic freeness principle, establishes that the spectrum of very general random matrices behaves as that of an associated free operator. This reduces the study of such random matrices to the deterministic problem of computing spectral statistics of the free operator. In the self-adjoint case, the spectral edges of the free operator can be computed exactly by means of a variational formula due to Lehner. In this note, we provide variational formulas for the largest and smallest singular values in the non-self-adjoint case.