TWO-SIDED MULTIPLICATION OPERATORS ON THE SPACE OF REGULAR OPERATORS.

Let W, X, Y and Z be Dedekind complete Riesz spaces. For A ε Lr(Y,Z) and B ε Lr(W,X) let MA,B be the two-sided multiplication operator from Lr(X, Y ) into Lr(W, Z) defined by MA,B(T) = ATB. We show that for every 0 ⩽ A0 ε Lrn (Y,Z), ǀMA,B,Bǀ(T) = MA0,ǀBǀ(T) holds for all B ε Lr(W,X) and all T ε Lrn(X, Y ). Furthermore, if W, X, Y and Z are Dedekind complete Banach lattices such that X and Y have order continuous norms, then ǀMA,Bǀ = MǀAǀ, ǀBǀ for all A ε Lr(Y,Z) and all B ε Lr(W,X). Our results generalize the related results of Synnatzschke and Wickstead, respectively. [ABSTRACT FROM AUTHOR]