| Title:
|
Commutants and derivation ranges (English) |
| Author:
|
Mecheri, Salah |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
49 |
| Issue:
|
4 |
| Year:
|
1999 |
| Pages:
|
843-847 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we obtain some results concerning the set ${\mathcal M} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal L(H)}\bigr \rbrace $, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal H$ stands for a Hilbert space and we prove that every compact operator in $\overline{R(\delta _A)}^w\cap \lbrace A^*\rbrace ^{\prime }$ is quasinilpotent if $A$ is dominant, where $\overline{R(\delta _A)}^w$ is the closure of the range of $\delta _A$ in the weak topology. (English) |
| MSC:
|
47A10 |
| MSC:
|
47A65 |
| MSC:
|
47B47 |
| idZBL:
|
Zbl 1008.47038 |
| idMR:
|
MR1746710 |
| . |
| Date available:
|
2009-09-24T10:28:23Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127534 |
| . |
| Reference:
|
[1] J.H.Anderson: On normal derivations.Proc. Amer. Math. Soc. 38 (1973), 135–140.. Zbl 0255.47036, MR 0312313, 10.1090/S0002-9939-1973-0312313-6 |
| Reference:
|
[2] J.H.Anderson: Derivation ranges and the identity.Bull. Amer. Math. Soc. 79 (1973), 705–708.. Zbl 0269.47021, MR 0322518, 10.1090/S0002-9904-1973-13271-9 |
| Reference:
|
[3] D.C.Kleïnecke: On operator commutators.Proc. Amer. Math. Soc. 8 (1957), 535–536.. Zbl 0079.12904, MR 0087914, 10.1090/S0002-9939-1957-0087914-4 |
| Reference:
|
[4] J.G.Stampfli, B.L.Wadhwa: On dominant operators.Monatsh. Math. 84 (1977), 143–153. Zbl 0374.47010, MR 0458225, 10.1007/BF01579599 |
| Reference:
|
[5] R.E.Weber: Derivations and the trace class operators.Proc. Amer. Math. Soc. 73 (1979), 79–82. Zbl 0372.47019, MR 0512062, 10.1090/S0002-9939-1979-0512062-0 |
| . |
