Normalized Orbit Frames and Essential Spectrum, II

Abstract

Given $0<r_j<1$ with $r_j\to 1$, there exist a normal operator $T$ on ${\ell }^2$ with eigenvalues ${\lambda }_j$ satisfying $|{\lambda }_j|=r_j$ and $g\in {\ell }^2$ such that \[ \left\{ \frac{T^{n}g}{\left\Vert T^{n}g\right\Vert }\right\} _{n\geq0}\text{is a frame}\:\Longleftrightarrow\:\sum_{j\geq1}\left(1-r^{2}_{j}\right)=\infty. \] Whenever this condition holds, the frame bounds can be chosen with ratio arbitrarily close to one.