| Title:
|
A new characterization of Anderson’s inequality in $C_1$-classes (English) |
| Author:
|
Mecheri, S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
697-703 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Cal H$ be a separable infinite dimensional complex Hilbert space, and let $\Cal L(\Cal H)$ denote the algebra of all bounded linear operators on $\Cal H$ into itself. Let $A=(A_{1},A_{2},\dots ,A_{n})$, $B=(B_{1},B_{2},\dots ,B_{n})$ be $n$-tuples of operators in $\Cal L(\Cal H)$; we define the elementary operators $\Delta_{A,B}\:\Cal L(\Cal H)\mapsto\Cal L(\Cal H)$ by $\Delta_{A,B}(X)=\sum_{i=1}^nA_iXB_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in\Cal L(\Cal H)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in\Cal L(\Cal H)$ such that $\sum_{i=1}^nB_iTA_i=T$ implies $\sum_{i=1}^nA_i^*TB_i^*=T$ for all $T\in\Cal C_1(\Cal H)$ (trace class operators). The main result is the equivalence between this property and the fact that the ultraweak closure of the range of the elementary operator $\Delta_{A,B}$ is closed under taking adjoints. This leads us to give a new characterization of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel in $C_1$ classes. (English) |
| Keyword:
|
$C_1$-class |
| Keyword:
|
generalized $p$-symmetric operator |
| Keyword:
|
Anderson Inequality |
| MSC:
|
47B20 |
| MSC:
|
47B47 |
| idZBL:
|
Zbl 1174.47025 |
| idMR:
|
MR2337624 |
| . |
| Date available:
|
2009-09-24T11:48:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128199 |
| . |
| Reference:
|
[1] J. H. Anderson: On normal derivation.Proc. Amer. Math. Soc. 38 (1973), 135–140. MR 0312313, 10.1090/S0002-9939-1973-0312313-6 |
| Reference:
|
[2] J. H. Anderson, J. W. Bunce, J. A. Deddens, and J. P. Williams: C$^{*}$ algebras and derivation ranges.Acta Sci. Math. 40 (1978), 211–227. MR 0515202 |
| Reference:
|
[3] S. Bouali, J. Charles: Extension de la notion d’opérateur D-symétique I.Acta Sci. Math. 58 (1993), 517–525. (French) MR 1264254 |
| Reference:
|
[4] S. Mecheri: Generalized P-symmetric operators.Proc. Roy. Irish Acad. 104A (2004), 173–175. MR 2140424 |
| Reference:
|
[5] S. Mecheri, M. Bounkhel: Some variants of Anderson’s inequality in $C_{1}$-classes.JIPAM, J. Inequal. Pure Appl. Math. 4 (2003), 1–6. MR 1966004 |
| Reference:
|
[6] S. Mecheri: On the range of elementary operators.Integral Equations Oper. Theory 53 (2005), 403–409. Zbl 1120.47024, MR 2186098, 10.1007/s00020-004-1327-3 |
| Reference:
|
[7] V. S. Shulman: On linear equation with normal coefficient.Dokl. Akad. Nauk USSR 2705 (1983), 1070–1073. (Russian) MR 0714059 |
| Reference:
|
[8] J. P. Williams: On the range of a derivation.Pac. J. Math. 38 (1971), 273–279. Zbl 0205.42102, MR 0308809 |
| . |
