TWO-SIDED MULTIPLICATION OPERATORS ON THE SPACE OF REGULAR OPERATORS

Let W, X, Y and Z be Dedekind complete Riesz spaces. For A is an element of L-r(Y, Z) and B is an element of L-r(W, X) let M-A,M- (B) be the two-sided multiplication operator from L-r(X, Y) into L-r(W, Z) defined by M-A,M- B(T) = ATB. We show that for every 0 <= A(0) is an element of L-n(r) (Y, Z), broken vertical bar M-A0,M- B broken vertical bar (T) = M-A0, (broken vertical bar B|) (T) holds for all B is an element of L-r(W, X) and all T is an element of L-n(r) (X, Y). Furthermore, if W, X, Y and Z are Dedekind complete Banach lattices such that X and Y have order continuous norms, then broken vertical bar M-A,M- B broken vertical bar = M broken vertical bar A broken vertical bar, |broken vertical bar B broken vertical bar for all A is an element of L-r(Y, Z) and all B is an element of L-r(W, X). Our results generalize the related results of Synnatzschke and Wickstead, respectively.