PERTURBATION OF BANACH SPACE OPERATORS WITH A COMPLEMENTED RANGE

Let${\mathcal C}[{\mathcal X}]$be any class of operators on a Banach space${\mathcal X}$, and let${Holo}^{-1}({\mathcal C})$denote the class of operatorsAfor which there exists a holomorphic functionfon a neighbourhood${\mathcal N}$of the spectrum σ(A) ofAsuch thatfis non-constant on connected components of${\mathcal N}$andf(A) lies in${\mathcal C}$. Let${{\mathcal R}[{\mathcal X}]}$denote the class of Riesz operators in${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators$A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$(the class of all upper or lower [semi] Fredholm operators) by commuting operators in$B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi= 1(B− μi) is Riesz, then there exist decompositions${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$and$B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$such that: (i) If λ ≠ 0, then$\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0)$A\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-B_0\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), whereB0= ⊕mi= 1(Bi− μi); (iii) if$\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then$A-\lambda B\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$).