Every nilpotent operator fails to determine the complete norm topology (Q2567453)

The investigation about operators determining the complete norm topology of a given Banach space \((X,\|\cdot\|)\) started in a paper authored by the reviewer [Stud. Math. 124, No. 2, 155--160 (1997; Zbl 0895.46015)] and it was continued independently by \textit{K. Jarosz} [Trans. Am. Math. Soc. 353, No. 2, 723--731 (2001; Zbl 0966.46007)] and the reviewer [Proc. Am. Math. Soc. 129, No. 4, 1057--1064 (2001; Zbl 0961.43004)]. The problem is to decide whether or not a given continuous linear operator \(T\) on \(X\) determines the given norm \(\|\cdot\|\) in the sense that every complete norm \(|\cdot|\) on \(X\) with the property that the operator \(T\colon(X,|\cdot|) \rightarrow(X,|\cdot|)\) is continuous is necessarily equivalent to \(\|\cdot\|\). In the paper under review, it is shown that every nilpotent operator on an infinite-dimensional Banach sapace \(X\) necessarily fails to determine the norm of \(X\).