# Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 593-603.

Spectral conditions for the nilpotency of Lie algebras**Authors**: Daniel Beltita

**Author institution:**Institute of Mathematics of the Romanian Academy, P.O. Box 1--764, 70700 Bucharest, Romania

**Summary:**In the paper it is proved that each of the following conditions below is suffici ent for the nilpotency of a solvable Lie algebra. (They are formulated by means of the spectrum of a Banach space representation o f a Lie algebra, introduced by C. Ott in [9].) 1) The validity of the projection property of the spectrum on Lie subalgebras (imposed to only one representation). 2) The existence of representations with spectra of arbitrarily small diameters (with respect to a fixed norm on the given Lie algebra).

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