Local properties of powers of operators (Q1894900)

Let \(T\) be a bounded operator on a Banach space \(X\), \(x\in X\) and \(\ell\in X^*\). Assume that \(|\langle T^ n x, \ell\rangle|= o(n^{k+ 1})(n\to\infty)\) for some integer \(k\geq 0\). Then either \(\limsup_{n\to\infty} n|\langle (T- I)^ n x,\ell\rangle|^{{1\over n}}> 0\), or \(\langle(T- I)^ n x, \ell\rangle= 0\) for all \(n\geq k+ 1\).