| Title:
|
Geometry of the spectral semidistance in Banach algebras (English) |
| Author:
|
Braatvedt, Gareth |
| Author:
|
Brits, Rudi |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
599-610 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A$ be a unital Banach algebra over $\mathbb C$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in \mathbb C$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if and only if all the Riesz projections, $p(\alpha ,a)$ and $p(\alpha ,b)$, correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space $X$ in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu's geometric formula (which requires the knowledge of the local spectra of the operators at each $0\not =x\in X$). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results. (English) |
| Keyword:
|
asymptotically intertwined |
| Keyword:
|
Riesz projection |
| Keyword:
|
spectral semidistance |
| Keyword:
|
quasinilpotent equivalent |
| MSC:
|
46H05 |
| MSC:
|
47A05 |
| MSC:
|
47A10 |
| idZBL:
|
Zbl 06391514 |
| idMR:
|
MR3298549 |
| DOI:
|
10.1007/s10587-014-0121-x |
| . |
| Date available:
|
2014-12-19T15:55:02Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144047 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Colojoară, I., Foiaş, C.: Quasi-nilpotent equivalence of not necessarily commuting operators.J. Math. Mech. 15 (1966), 521-540. MR 0192344 |
| Reference:
|
[4] Foiaş, C., Vasilescu, F.-H.: On the spectral theory of commutators.J. Math. Anal. Appl. 31 (1970), 473-486. MR 0290146, 10.1016/0022-247X(70)90001-6 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[7] Razpet, M.: The quasinilpotent equivalence in Banach algebras.J. Math. Anal. Appl. 166 (1992), 378-385. Zbl 0802.46064, MR 1160933, 10.1016/0022-247X(92)90304-V |
| Reference:
|
[8] Vasilescu, F.-H.: Analytic Functional Calculus and Spectral Decompositionsk.Mathematics and Its Applications (East European Series) 1 D. Reidel Publishing, Dordrecht (1982), translated from the Romanian. MR 0690957 |
| Reference:
|
[9] Vasilescu, F.-H.: Some properties of the commutator of two operators.J. Math. Anal. Appl. 23 (1968), 440-446. Zbl 0159.43402, MR 0229078, 10.1016/0022-247X(68)90080-2 |
| Reference:
|
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| . |
