A note on a result by M. Oudghiri and K. Souilah

Abstract

M. Oudghiri and K. Souilah recently proved that if a bounded linear
operator R ∈ L(X) is Riesz, then the following conditions are equivalent:
(i) R = Q + F, where Q ∈ L(X) is quasinilpotent, F ∈ L(X)
is finite rank and QF = FQ; (ii) σgD(T) = σgD(T + R) for every
T ∈ L(X) commuting with R, where σgD(·) denotes the generalized
Drazin spectrum. In this note we present two new conditions that are
equivalent to the previously mentioned conditions (i) and (ii). Additionally,
we offer an alternative proof for the implication (i) ⇒ (ii) of
the above result.