Two-sided multiplication operators on the space of regular operators

TL;DR

This paper investigates the properties of two-sided multiplication operators on spaces of regular operators within Dedekind complete Riesz and Banach lattices, establishing conditions under which their modulus can be expressed as a multiplication of moduli.

Contribution

It generalizes existing results by characterizing the modulus of two-sided multiplication operators in more general lattice settings.

Findings

For every positive regular operator, the modulus of the multiplication operator equals the multiplication of the modulus operators.

Under certain conditions, the modulus of the multiplication operator equals the multiplication of the moduli of the individual operators.

The results extend previous work by Synnatzschke and Wickstead to broader classes of lattices.

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 $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. 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.