A note on the index of $B$-Fredholm operators

Berkani, M., and Medková, Dagmar. "A note on the index of $B$-Fredholm operators." Mathematica Bohemica 129.2 (2004): 177-180. <http://eudml.org/doc/249390>.

@article{Berkani2004,
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 $\operatorname\{\text\{ind\}\}(ST)= \operatorname\{\text\{ind\}\}(S) +\operatorname\{\text\{ind\}\}(T)$, 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.},
author = {Berkani, M., Medková, Dagmar},
journal = {Mathematica Bohemica},
keywords = {$B$-Fredholm operators; index of the product of Fredholm operators; -Fredholm operators; index of the product of Fredholm operators},
language = {eng},
number = {2},
pages = {177-180},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the index of $B$-Fredholm operators},
url = {http://eudml.org/doc/249390},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Berkani, M.
AU - Medková, Dagmar
TI - A note on the index of $B$-Fredholm operators
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 2
SP - 177
EP - 180
AB - 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.
LA - eng
KW - $B$-Fredholm operators; index of the product of Fredholm operators; -Fredholm operators; index of the product of Fredholm operators
UR - http://eudml.org/doc/249390
ER -