On the essential approximate point spectrum of operators (Q1803935)

In connection with the invariant subspace problem, the author shows that there exists an \(x\) in \(X\) such that \(\| p(T)x\|\geq a_{\deg p}| p(\lambda)|\) for every polynomial \(p\), if \(T\) is an operator on a Banach space \(X\), \(\lambda\) is in the essential approximate point spectrum of \(T\) (i.e. \(T-\lambda\) is not upper semi-Fredholm), and \(a_ 0,a_ 1,a_ 2,\ldots\) is a sequence of positive numbers converging to 0. In addition, he remarks that it is impossible to replace \(a_{\deg p}\) in the above inequality by any constant \(c>0\), unless \(T\) has a nontrivial invariant subspace.