On nilpotent operators
Volume 166 / 2005
Abstract
We give several necessary and sufficient conditions in order that a bounded linear operator on a Banach space be nilpotent. We also discuss three necessary conditions for nilpotency. Furthermore, we construct an infinite family (in one-to-one correspondence with the square-summable sequences $(\varepsilon _n)_{n\in {\mathbb N}}$ of strictly positive real numbers) of nonnilpotent quasinilpotent operators on an infinite-dimensional Hilbert space, all the iterates of each of which have closed range. Each of these operators (as well as an operator previously constructed by C. Apostol in [Ap]) can be used to provide a negative answer to a question posed by M. Mbekhta and J. Zemánek [MZ]. We also use our example to show that two (equivalent to each other) of the three necessary conditions for nilpotency we have mentioned above are not sufficient, by proving that the sequence $(\varepsilon _n)_{n\in {\mathbb N}}$ can be chosen so that these two conditions are satisfied. Finally, from a generalization—obtained by using a theorem proved by M. Gonzalez and V. M. Onieva in [GO2]—of a result provided by C. Apostol in [Ap], we derive that any holomorphic function of each operator in our example, as well as of the one constructed in [Ap], has closed range.
