| Title:
|
A note on the index of $B$-Fredholm operators (English) |
| Author:
|
Berkani, M. |
| Author:
|
Medková, D. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
129 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
177-180 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$B$-Fredholm operators |
| Keyword:
|
index of the product of Fredholm operators |
| MSC:
|
47A53 |
| MSC:
|
47A55 |
| idZBL:
|
Zbl 1056.47011 |
| idMR:
|
MR2073513 |
| DOI:
|
10.21136/MB.2004.133905 |
| . |
| Date available:
|
2009-09-24T22:14:06Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133905 |
| . |
| Reference:
|
[1] Berkani, M.: On a class of quasi-Fredholm operators.Integral Equations Oper. Theory 34 (1999), 244–249. Zbl 0939.47010, MR 1694711, 10.1007/BF01236475 |
| Reference:
|
[2] Berkani, M.: Restriction of an operator to the range of its powers.Stud. Math. 140 (2000), 163–175. Zbl 0978.47011, MR 1784630, 10.4064/sm-140-2-163-175 |
| Reference:
|
[3] Berkani, M.: Index of $B$-Fredholm operators and generalization of a Weyl Theorem.Proc. Amer. Math. Soc. 130 (2002), 1717–1723. Zbl 0996.47015, MR 1887019, 10.1090/S0002-9939-01-06291-8 |
| Reference:
|
[4] Berkani, M.; Sarih, M.: On semi $B$-Fredholm operators.Glasg. Math. J. 43 (2001), 457–465. MR 1878588, 10.1017/S0017089501030075 |
| Reference:
|
[5] Berkani, M. ; Sarih, M.: An Atkinson-type theorem for $B$-Fredholm operators.Stud. Math. 148 (2001), 251–257. MR 1880725, 10.4064/sm148-3-4 |
| Reference:
|
[6] Grabiner, S.: Uniform ascent and descent of bounded operators.J. Math. Soc. Japan 34 (1982), 317–337. Zbl 0477.47013, MR 0651274, 10.2969/jmsj/03420317 |
| Reference:
|
[7] Heuser, H.: Funktionalanalysis.Teubner, Stuttgart, 1975. Zbl 0309.47001, MR 0482021 |
| Reference:
|
[8] Kordula, V.; Müller, V.: On the axiomatic theory of the spectrum.Stud. Math. 119 (1996), 109–128. MR 1391471 |
| Reference:
|
[9] Laursen, K. B.; Neumann, M. M.: An Introduction to Local Spectral Theory.Clarendon Press, Oxford, 2000. MR 1747914 |
| Reference:
|
[10] Mbekhta, M.; Müller, V.: On the axiomatic theory of the spectrum, II.Stud. Math. 119 (1996), 129–147. MR 1391472, 10.4064/sm-119-2-129-147 |
| . |
