Lower-shift displacement rank and generator pairs

For $A\in {\mathbb{C}}^{n\times n}$, $n\ge 1$, let $Z=Z_n$ be the lower shift with $Z_{i+1,i}=1$ for $0\le i<n-1$ and all other entries zero. Its lower-shift (Stein) displacement is ${\mathrm{\nabla }}_Z(A):=A-ZA{Z}^{\ast }\mathrm{.}$ The displacement rank of $A$ with respect to this fixed shift is $\mathrm{rank}({\mathrm{\nabla }}_Z(A))$. A pair of matrices $G,H\in {\mathbb{C}}^{n\times r}$ is a generator of width $r$ when ${\mathrm{\nabla }}_Z(A)=G{H}^{\ast }\mathrm{.}$ Such a factorization is a representation of the displacement, not a factorization of $A$ itself. The term Toeplitz-like refers to matrices whose displacement rank is small relative to $n$; the rank bound $r$ must be specified rather than unrestricted. For a family of growing dimensions, a uniform bound means that $r$ is independent of $n$. This definition makes no conditioning, pivot-growth, or numerical-stability assertion.