| Title:
|
Subsets of nonempty joint spectrum in topological algebras (English) |
| Author:
|
Wawrzyńczyk, Antoni |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
143 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
441-448 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $\mathcal A$ to have a non-void left joint spectrum $\sigma _l(S).$ In particular, for a Lie subalgebra $L\subset \mathcal A$ we have $\sigma _l(L)\neq \emptyset $ if and only if $[L,L]$ generates in $\mathcal A$ a proper left ideal. \endgraf We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid. (English) |
| Keyword:
|
joint spectrum |
| Keyword:
|
Waelbroeck algebra |
| Keyword:
|
commutator |
| Keyword:
|
spectral mapping formula |
| MSC:
|
46H30 |
| MSC:
|
47A13 |
| MSC:
|
47A60 |
| idZBL:
|
Zbl 06997377 |
| idMR:
|
MR3895267 |
| DOI:
|
10.21136/MB.2018.0098-17 |
| . |
| Date available:
|
2018-11-29T09:26:33Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147480 |
| . |
| Reference:
|
[1] Harte, R. E.: Spectral mapping theorem.Proc. R. Ir. Acad., Sect. A 72 (1972), 89-107. Zbl 0206.13301, MR 0326394 |
| Reference:
|
[2] Janas, J.: Note on the joint spectrum of the Wiener-Hopf operators.Proc. Am. Math. Soc. 50 (1975), 303-308. Zbl 0337.47017, MR 0374977, 10.2307/2040557 |
| Reference:
|
[3] Müller, V., So{ł}tysiak, A.: Spectrum of generators of a noncommutative Banach algebra.Studia Math. 93 (1989), 87-95. Zbl 0704.46027, MR 0989569, 10.4064/sm-93-1-87-95 |
| Reference:
|
[4] Nuñez, J. R.: A Joint Spectrum Associated to an Ideal.Tesis de Maestría, Universidad Autónoma Metropolitana, Ciudad de México (2017). |
| Reference:
|
[5] Pryde, A. J., So{ł}tysiak, A.: On joint spectra of non-commuting normal operators.Bull. Aust. Math. Soc. 48 (1993), 163-170. Zbl 0810.47003, MR 1227446, 10.1017/S0004972700015562 |
| Reference:
|
[6] Wawrzyńczyk, A.: Joint spectra in Waelbroeck algebras.Bol. Soc. Mat. Mex., III. Ser. 13 (2007), 321-343. Zbl 1178.46047, MR 2472509 |
| Reference:
|
[7] Wawrzyńczyk, A.: Schur lemma and the spectral mapping formula.Bull. Pol. Acad. Sci., Math. 55 (2007), 63-69. Zbl 1118.46045, MR 2304300, 10.4064/ba55-1-7 |
| . |
