**
Czechoslovak Mathematical Journal, Vol. 57, No. 2, pp. 697-703, 2007
**

#
A new characterization of Anderson's Inequality

in $C_1$-classes

##
S. Mecheri

* Salah Mecheri*, Department of Mathematics, King Saud University, College of Science, P.O. Box 2455, Riyadh 11451, Saudi Arabia, e-mail: ` mecherisalah@hotmail.com`

**Abstract:** Let ${\Cal H}$ be a separable infinite dimensional complex Hilbert space, and let ${\Cal L(H)}$ 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 ${\Cal L(H)}$; we define the elementary operators $\Delta_{A,B} {\Cal L(H)}\mapsto{\Cal L(H)}$ by

\Delta_{A,B}(X)=\sum_{i=1}^nA_iXB_i-X.

In this paper, we characterize the class of pairs of operators $A,B\in{\Cal L(H)}$ satisfying Putnam-Fuglede's property, i.e, the class of pairs of operators $A,B\in{\Cal L(H)}$ 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.

**Keywords:** $C_1$-class, generalized $p$-symmetric operator, Anderson Inequality

**Classification (MSC 2000):** 47B47, 47B20

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