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$C_1$-class; generalized $p$-symmetric operator; Anderson Inequality
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.
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