This is an announcement for the paper “Two-sided multiplication operators on the space of regular operators” by Jin Xi Chenhttps://arxiv.org/find/math/1/au:+Chen_J/0/1/0/all/0/1, Anton R. Schephttps://arxiv.org/find/math/1/au:+Schep_A/0/1/0/all/0/1.
Abstract: Let $W, X, Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^r(Y, Z)$ and $B\in L^r(W, X)$ let $M_{A, B}$ be the two-sided multiplication operator from $L^r(X, Y)$ into $L^r(W, Z)$ defined by $M_{A, B}(T)=ATB$. We show that for every $0\leq A_0\in L^{rn}(Y, Z), |M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^r(W, X)$ and all $T\in L^{rn}(X, Y)$. Furthermore, if $W, X, Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A, B}|=M_{|A|, |B|}$ for all $A\in L^r(Y, Z)$and all $B\in L^r(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.06913