(1) Given operators A>0 and B>0 on a space , the following statements are equivalent: (i) A B; (ii) there exists an operator Z>0 on a space such that and ; (iii) there exists an operator Z>0 on a space such that and . (2) For p=−1,2,3 we show that the condition 0
0 on such that and . Several related results are proved and open problems are considered.