On the unicity of the Kato decomposition

The aim of this work is to study the unicity of the Kato decomposition associated to a pair of operators. We will prove that this decomposition is unique up to isomorphisms and we will give necessary and sufficient conditions to have unicity. This will generalize some results due to Mbekhta and Ouahab (Acta Sci. Math. (Szeged) 59 (1994) 525–543).  2003 Elsevier Inc. All rights reserved.