Generalization of some commutative perturbation results

Abstract

We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results. IntroductionIn [6], the authors investigated how to explicitly express the Drazin inverse of the sum (P + Q) of two complex matrices P and Q, under the conditions P Q ∈ comm(P ) and QP ∈ comm(Q), which are weaker than the commutativity of P with Q. A few years later, Huihui Zhu et al. [13] obtained the representations for the pseudo Drazin inverse of the sum and the product of two elements of a complex Banach algebra, under the same conditions. Recently, H. Zou et al. [14] extended the known expressions for the generalized Drazin inverse of the product and the sum of two elements of a complex Banach algebra by considering the same conditions. In this paper, we study these conditions and other in a ring A, that are ab ∈ comm(a), ba ∈ comm(b), ab ∈ comm(b) and ba ∈ comm(a). After giving some algebraic results, we focus on the Banach algebra of bounded linear operators L(X) acting on the complex Banach algebra X. We generalize some commutative perturbation spectral results, in particular, if N is nilpotent and N ∈ comm r (T ) (i.e. N T ∈ comm(T ) and T N ∈ comm(N )), then σ * (T ) \ {0} = σ * (T + N ) \ {0}, where σ * ∈ {σ p , σ 0 p , σ a }. If in addition N ∈ comm w (T ) (i.e. N ∈ comm r (T ) and N * ∈ comm r (T * )), then we deduce by duality that σ * (T ) = σ * (T + N ), where σ * ∈ {σ p , σ 0 p , σ a , σ s , σ}. This allows us to show that if K is a power compact operator and K ∈ comm w (T ), then σ * (T ) = σ * (T + K), where σ * ∈ {σ uf , σ uw , σ ub , σ e , σ w , σ b }. Note that here we lose quite a few commutative properties, for example if S ∈ comm(T ) and S ∈ comm w (T ), then S ∈ comm w (T − λI) for every λ = 0. Terminology and preliminariesLet A be a ring and let a ∈ A. Denote by comm(a) the set of all elements that commute with a, by comm 2 (a) = comm(comm(a)) and by Nil(A) the nilradical of A. If in addition A is a complex Banach algebra with unit e, then we means by σ(a), acc σ(a), r(a) and exp(a), the spectrum, the accumulation point of σ(a), the spectral radius of a and the exponential of a, respectively. We