A lower bound for the differences of powers of linear operators (Q2641539)

The authors prove that, if \(L\) is a quasi-nilpotent bounded linear operator in a Banach space \(X\) such that \(L \neq 0\), then \[ \liminf_{n\to\infty}(n+1)\|(I-L)^nL\|\geq {\frac{1}{e}}. \] They also prove that, for any \(T\in {\mathcal L}(X)\), either (i) \(\limsup_{n\to\infty}(n+1)\|(I-T)T^n\|\geq \frac{1}{e}\), or (ii) \(\limsup_{n\to\infty}(n+1)\|(I-T)T^n\|=0\).