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

Mecheri, 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 -