M. Berkani, Groupe d'Analyse et Theorie des Operateurs (G.A.T.O), Universite Mohammed I, Faculte des Sciences, Departement de Mathematiques, Oujda, Maroc, e-mail: berkani@sciences.univ-oujda.ac.ma; D. Medkova, Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1; Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nam. 13, Praha 2, Czech Republic, e-mail: medkova@math.cas.cz
Abstract: From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251-257] we know that if $S$, $ T$ are commuting $B$-Fredholm operators acting on a Banach space $X$, then $ST$ is a $B$-Fredholm operator. In this note we show that in general we do not have $\ind(ST)= \ind(S) +\ind(T)$, contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717-1723]. However, if there exist $ U, V \in L(X) $ such that $S$, $T$, $U$, $V$ are commuting and $ US+ VT= I$, then $\ind(ST)= \ind(S)+\ind(T)$, where $\ind$ stands for the index of a $B$-Fredholm operator.
Keywords: $B$-Fredholm operators, index
Classification (MSC 2000): 47A53, 47A55
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