Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant rand