arXiv:1505.02719v2 [math.FA] 25 Feb 2017 RIGHT (OR LEFT) INVERTIBILITY OF BOUNDED AND UNBOUNDED OPERATORS AND APPLICATIONS TO THE SPECTRUM OF PRODUCTS SOUHEYB DEHIMI AND MOHAMMED HICHEM MORTAD * Abstract. This paper is mainly concerned with proving σ(AB)= σ(BA) for two linear and non necessarily bounded operators A and B. The main tool is left and right invertibility of bounded and unbounded operators. 1. Introduction All operators considered here are linear and defined on a complex separable Hilbert space H . In order to avoid trivialities in the bounded case, we further assume that dim H = ∞. Also, we assume that the reader is well aware of the basic notions of bounded and unbounded operators as well as the algebraic notions of right and left invertibility. It is known that if no condition is imposed on either of the operators A or B, then we are only sure that: σ(AB) −{0} = σ(BA) −{0}. We would like to know when σ(AB)= σ(BA) ······ (E) holds for two linear bounded operators. We already know that if one of the operators is invertible, then it may be shown that AB and BA are similar, hence they have the same spectrum. (E) is also satisfied when one of the operators is compact. Hladnik-Omladič [5] proved the following: Theorem 1.1. Let A, B ∈ B(H ) be such that B is positive. Let P be the (unique) square root of B. Then σ(AB)= σ(BA)= σ(P AP ). Another case for which the equality σ(AB)= σ(BA) holds is when one of the operators is normal: Theorem 1.2. (Barraa-Boumazghour, [1]) Let A, B ∈ B(H ) be such that one of them is normal. Then σ(AB)= σ(BA). 2010 Mathematics Subject Classification. Primary 47A05, Secondary 47A10, 47B20, 47B25. Key words and phrases. Normal and Self-adjoint Operators. Right Invertibility, Left Invertibility. Invertibility. Spectrum of Products. * Corresponding author. 1