On nilpotent operators

Laura Burlando¹

¹ Dipartimento di Matematica dell'Università di Genova Via Dodecaneso 35 16146 Genova, Italy

Studia Mathematica, Tome 166 (2005) no. 2, pp. 101-129

Résumé

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.

Détail

DOI : 10.4064/sm166-2-1

Keywords: several necessary sufficient conditions order bounded linear operator banach space nilpotent discuss three necessary conditions nilpotency furthermore construct infinite family one to one correspondence square summable sequences varepsilon mathbb strictly positive real numbers nonnilpotent quasinilpotent operators infinite dimensional hilbert space iterates each which have closed range each these operators operator previously constructed apostol provide negative answer question posed mbekhta zem nek example equivalent each other three necessary conditions nilpotency have mentioned above sufficient proving sequence varepsilon mathbb chosen these conditions satisfied finally generalization obtained using theorem proved gonzalez onieva result provided apostol derive holomorphic function each operator example constructed has closed range