Comment on Maps preserving the dimension of fixed points of products of operators

Let $\mathcal{B} (X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. In this note, we show that a lemma used in the proof of the main result of [ Taghavi and Hosseinzadeh, linear and Multilinear algebra (2013) 1285-1292.] has an incorrect proof. Then, instead of such a lemma, we provide two elementary lemmas to obtain a proof of Taghavi and Hosseinzadeh's main result free of errors.