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 π
B
(B) = ∏
m
i = 1(B − μ
i
) 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
− μ
i
); (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.,
$\in\Phi_{-}({\mathcal X})$
).