ON THE PRODUCT OF DERIVATIONS IN BANACH ALGEBRAS Nadia Boudi D´ epartement de Math´ ematiques, Facult´ e des Sciences, Universit´ e Moulay Ismail, Meknes, Morocco [Accepted 2 June 2009. Published 11 September 2009.] Dedicated to Professor El Amin Kaidi on the occasion of his sixtieth birthday Abstract The purpose of this paper is to study derivations d, g defined on a complex Banach algebra A such that the spectrum σ(dg(x)) is finite for all x ∈ A. In particular we show that if the algebra A is semisimple, then dg(x) 3 lies in the socle of A for every x ∈ A. 1. Introduction In [8], Breˇ sar and ˇ Semrl studied commuting pairs d and g of continuous derivations defined on a Banach algebra A such that dg(x) is quasi-nilpotent for all x ∈ A. A short proof of this result can be found in [13]. In [9], Chebotar, Ke and Lee showed that the assumption of commutativity dg = gd is unnecessary. The reader is referred to [8] for the general motivation of this research direction. The present paper continues this line of investigations. Our goal is to show that if d, g are con- tinuous derivations defined on a complex Banach algebra A such that the spectrum σ(dg(x)) is finite for all x ∈ A, then (dg(x)) 3 lies in the socle modulo the radical for every x ∈ A. It should be pointed out that our study is closely connected with questions concerning derivations mapping into the socle [5; 6; 7]. 2. The case of dense algebras Let A be an algebra. By [x, y] we denote the commutator xy − yx of x, y ∈ A. For each x ∈ A let δ x denote the inner derivation δ x (y)=[x, y] implemented by x. If A is semisimple, the socle soc A of A is defined as the sum of all minimal left ideals of A. If there are no minimal left ideals, then soc A = 0 by definition. The Jacobson radical of A will be denoted by rad A. Let X be a complex vector space * Corresponding author, e-mail: nadia boudi@hotmail.com 2000 Mathematics Subject Classification:47B47, 47B48, 47A10. doi:10.3318/PRIA.2009.109.2.201 Cite as follows: Nadia Boudi, On the product of derivations in Banach algebras, Mathematical Proceedings of the Royal Irish Academy 109A (2009), 201–211; doi:10.3318/PRIA.2009.109.2.201. Mathematical Proceedings of the Royal Irish Academy, 109A (2), 201–211 (2009) c  Royal Irish Academy