# A new characterization of Anderson’s inequality in ${C}_{1}$-classes

Czechoslovak Mathematical Journal (2007)

- Volume: 57, Issue: 2, page 697-703
- ISSN: 0011-4642

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topMecheri, S.. "A new characterization of Anderson’s inequality in $C_1$-classes." Czechoslovak Mathematical Journal 57.2 (2007): 697-703. <http://eudml.org/doc/31156>.

@article{Mecheri2007,

abstract = {Let $\mathcal \{H\}$ be a separable infinite dimensional complex Hilbert space, and let $\mathcal \{L\}(\mathcal \{H\})$ denote the algebra of all bounded linear operators on $\mathcal \{H\}$ 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 $\mathcal \{L\}(\mathcal \{H\})$; we define the elementary operators $\Delta _\{A,B\}\:\mathcal \{L\}(\mathcal \{H\})\mapsto \mathcal \{L\}(\mathcal \{H\})$ by $\Delta _\{A,B\}(X)=\sum _\{i=1\}^nA_iXB_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in \mathcal \{L\}(\mathcal \{H\})$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in \mathcal \{L\}(\mathcal \{H\})$ such that $\sum _\{i=1\}^nB_iTA_i=T$ implies $\sum _\{i=1\}^nA_i^*TB_i^*=T$ for all $T\in \mathcal \{C\}_1(\mathcal \{H\})$ (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.},

author = {Mecheri, S.},

journal = {Czechoslovak Mathematical Journal},

keywords = {$C_1$-class; generalized $p$-symmetric operator; Anderson Inequality; -class; generalized -symmetric operator; Anderson inequality},

language = {eng},

number = {2},

pages = {697-703},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A new characterization of Anderson’s inequality in $C_1$-classes},

url = {http://eudml.org/doc/31156},

volume = {57},

year = {2007},

}

TY - JOUR

AU - Mecheri, S.

TI - A new characterization of Anderson’s inequality in $C_1$-classes

JO - Czechoslovak Mathematical Journal

PY - 2007

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 57

IS - 2

SP - 697

EP - 703

AB - Let $\mathcal {H}$ be a separable infinite dimensional complex Hilbert space, and let $\mathcal {L}(\mathcal {H})$ denote the algebra of all bounded linear operators on $\mathcal {H}$ 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 $\mathcal {L}(\mathcal {H})$; we define the elementary operators $\Delta _{A,B}\:\mathcal {L}(\mathcal {H})\mapsto \mathcal {L}(\mathcal {H})$ by $\Delta _{A,B}(X)=\sum _{i=1}^nA_iXB_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in \mathcal {L}(\mathcal {H})$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in \mathcal {L}(\mathcal {H})$ such that $\sum _{i=1}^nB_iTA_i=T$ implies $\sum _{i=1}^nA_i^*TB_i^*=T$ for all $T\in \mathcal {C}_1(\mathcal {H})$ (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.

LA - eng

KW - $C_1$-class; generalized $p$-symmetric operator; Anderson Inequality; -class; generalized -symmetric operator; Anderson inequality

UR - http://eudml.org/doc/31156

ER -

## References

top- 10.1090/S0002-9939-1973-0312313-6, Proc. Amer. Math. Soc. 38 (1973), 135–140. (1973) MR0312313DOI10.1090/S0002-9939-1973-0312313-6
- C${}^{*}$ algebras and derivation ranges, Acta Sci. Math. 40 (1978), 211–227. (1978) MR0515202
- Extension de la notion d’opérateur D-symétique I, Acta Sci. Math. 58 (1993), 517–525. (French) (1993) MR1264254
- Generalized P-symmetric operators, Proc. Roy. Irish Acad. 104A (2004), 173–175. (2004) MR2140424
- Some variants of Anderson’s inequality in ${C}_{1}$-classes, JIPAM, J. Inequal. Pure Appl. Math. 4 (2003), 1–6. (2003) MR1966004
- 10.1007/s00020-004-1327-3, Integral Equations Oper. Theory 53 (2005), 403–409. (2005) Zbl1120.47024MR2186098DOI10.1007/s00020-004-1327-3
- On linear equation with normal coefficient, Dokl. Akad. Nauk USSR 2705 (1983), 1070–1073. (Russian) (1983) MR0714059
- On the range of a derivation, Pac. J. Math. 38 (1971), 273–279. (1971) Zbl0205.42102MR0308809

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