On the spectra of products and linear combinations of idempotents
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Tusi Mathematical Research Group
ORIGINAL PAPER
On the spectra of products and linear combinations of idempotents Mohamed Barraa1 • El Hassan Benabdi1 • Mohamed Boumazgour2 Received: 25 August 2020 / Accepted: 16 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract Let A be a unital complex Banach algebra and let p; q 2 A be two idempotents. In this paper, we study the relationships between different spectra of pq and those of ap þ bq cpq, where a; b; c 2 C. Keywords Banach algebra Spectrum Drazin spectrum Idempotent
Mathematics Subject Classification 47A10 47A53
1 Introduction Let A be a unital complex Banach algebra with unit e. For a 2 A, let rðaÞ, rl ðaÞ and rr ðaÞ denote the spectrum, the left spectrum and the right spectrum of a, respectively. An element a 2 A is called idempotent if a2 ¼ a; it is called generalized Drazin invertible, if there exists b 2 A such that ab ¼ ba, bab ¼ b and a aba is quasinilpotent. If, in particular, a aba is nilpotent, then a is said to be Drazin invertible. Communicated by Pietro Aiena. & El Hassan Benabdi [email protected] Mohamed Barraa [email protected] Mohamed Boumazgour [email protected] 1
Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P.O.B: 2390, Marrakesh, Morocco
2
Faculty of Economical Science, Ibn Zohr University, P.O.B: 8658, Dakhla City, Agadir 80000, Morocco
M. Barraa et al.
The Drazin spectrum and the generalized Drazin spectrum of a 2 A are defined by rD ðaÞ ¼ fk 2 C : ke a is not Drazin invertibleg and rgD ðaÞ ¼ fk 2 C : ke a is not generalized Drazin invertibleg; respectively. The present paper is devoted to the study of the relationships between the spectrum, the left spectrum and the right spectrum of the product and a linear combination of two idempotents in A, respectively. The relationships between the Drazin and the generalized Drazin spectra of these elements are also investigated. In the last section, we give relationships between different spectra of the product PQ and those of aP þ bQ, where P and Q are bounded linear idempotent operators on a complex Banach space and a; b 2 C n f0g. The invertibility of a linear combination of two idempotents p; q 2 A has been studied in many papers, see [2, 4–6] and the references therein. In [3], the authors established relations between rðp þ qÞ, rðp qÞ and rðpqÞ, however the lemma they used to prove their main result fails to be true. In [6], a different proof is used to give a relationship between rðpqÞ and rðap þ bqÞ, where a; b 2 C n f0g. In the second section of this paper, we establish a relationship between the spectrum of pq and the one of ap þ bq cpq, where a; b; c 2 C; then we give a correction of the proof of [3, Theorem 1]. Section 3 is devoted to the study of the relationships between Drazin spectra and generalized Drazin spectra of pq and ap þ bq, where a; b 2 C n f0g, respectively. In the last section, we give some applications of the results obtained to the
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