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

TL;DR

This paper corrects an error in a previous proof regarding maps that preserve fixed point dimensions of operator products, providing a new, error-free proof with elementary lemmas.

Contribution

It identifies a flaw in prior work and offers a simplified, correct proof for maps preserving fixed point dimensions of operator products.

Findings

Corrected the proof of a main result in operator theory.

Provided two elementary lemmas to replace an incorrect lemma.

Established a valid proof for maps preserving fixed point dimensions.

Abstract

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.