# Article

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Keywords:
$B$-Fredholm operators; index of the product of Fredholm operators
Summary:
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 $\operatorname{\text{ind}}(ST)= \operatorname{\text{ind}}(S) +\operatorname{\text{ind}}(T)$, 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 $\operatorname{\text{ind}}(ST)= \operatorname{\text{ind}}(S)+\operatorname{\text{ind}}(T)$, where $\operatorname{\text{ind}}$ stands for the index of a $B$-Fredholm operator.
References:
[1] Berkani, M.: On a class of quasi-Fredholm operators. Integral Equations Oper. Theory 34 (1999), 244–249. DOI 10.1007/BF01236475 | MR 1694711 | Zbl 0939.47010
[2] Berkani, M.: Restriction of an operator to the range of its powers. Stud. Math. 140 (2000), 163–175. DOI 10.4064/sm-140-2-163-175 | MR 1784630 | Zbl 0978.47011
[3] Berkani, M.: Index of $B$-Fredholm operators and generalization of a Weyl Theorem. Proc. Amer. Math. Soc. 130 (2002), 1717–1723. DOI 10.1090/S0002-9939-01-06291-8 | MR 1887019 | Zbl 0996.47015
[4] Berkani, M.; Sarih, M.: On semi $B$-Fredholm operators. Glasg. Math. J. 43 (2001), 457–465. DOI 10.1017/S0017089501030075 | MR 1878588
[5] Berkani, M. ; Sarih, M.: An Atkinson-type theorem for $B$-Fredholm operators. Stud. Math. 148 (2001), 251–257. DOI 10.4064/sm148-3-4 | MR 1880725
[6] Grabiner, S.: Uniform ascent and descent of bounded operators. J. Math. Soc. Japan 34 (1982), 317–337. DOI 10.2969/jmsj/03420317 | MR 0651274 | Zbl 0477.47013
[7] Heuser, H.: Funktionalanalysis. Teubner, Stuttgart, 1975. MR 0482021 | Zbl 0309.47001
[8] Kordula, V.; Müller, V.: On the axiomatic theory of the spectrum. Stud. Math. 119 (1996), 109–128. MR 1391471
[9] Laursen, K. B.; Neumann, M. M.: An Introduction to Local Spectral Theory. Clarendon Press, Oxford, 2000. MR 1747914
[10] Mbekhta, M.; Müller, V.: On the axiomatic theory of the spectrum, II. Stud. Math. 119 (1996), 129–147. DOI 10.4064/sm-119-2-129-147 | MR 1391472

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