Normalized Orbit Frames and Essential Spectrum, II

Given $0<r_j<1$ with $r_j\to 1$, there exist a normal operator $T$ on ${\mathrm{\ell }}^2$ with eigenvalues ${\lambda }_j$ satisfying $\mathrm{\mid }{\lambda }_j\mathrm{\mid }=r_j$ and $g\in {\mathrm{\ell }}^2$ such that $${\left\{\frac{T^{n}g}{\parallel T^{n}g\parallel }\right\}}_{n\ge 0}\text{is a frame}⟺\sum _{j\ge 1}(1-r_j^2)=\mathrm{\infty }\mathrm{.}$$ Whenever this condition holds, the frame bounds can be chosen with ratio arbitrarily close to one.