PERTURBATION OF BANACH SPACE OPERATORS WITH A COMPLEMENTED RANGE

DUGGAL, B. P. ; KUBRUSLY, C. S.

Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 659-671

Résumé

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 operators A for which there exists a holomorphic function f on a neighbourhood ${\mathcal N}$ of the spectrum σ(A) of A such that f is non-constant on connected components of ${\mathcal N}$ and f(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) = ∏m i = 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., $\in\Phi_{-}({\mathcal X})$ ), where B 0 = ⊕m i = 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., $\in\Phi_{-}({\mathcal X})$ ).

Détail

DOI : 10.1017/S001708951600046X