Can we upper bound the convergence rate of $$\max_{\textbf{v}: \left\Vert \textbf{v}\right\Vert_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\right\Vert^2_2 \right\}~,$$ where $\textbf{T}\in \mathbb{R}^{d \times d}$ is a contraction operator ($\left\Vert\textbf{T}\right\Vert_2\le1$) of rank $r<d$?

For example, $\textbf{T}$ can be *nilpotent* with index $q$ (e.g., $\left[\begin{array}{cc}
0 & 1\\
0 & 0
\end{array}\right]
$) and then the maximal decrease *can* be fixed as long as $n<q$, and afterwards it is $0$.

I found a Toeplitz operator whose rate is $d/en$: $$\boldsymbol{T}=\left[\begin{array}{ccccc} & 1\\ & & \ddots\\ & & & 1\\ 1-\epsilon \end{array}\right]$$ and I have an intuition this should be the upper bound, but I was not able to prove that this is indeed the worst case.

8more comments