Abstract. In the physical papers [1]--[4] there are some interesting propositions about operators acting in Hilbert spaces. For example in [3] there were considered all bijective mappings on ${\cal B}_s(H)$, which preserve the order in both directions. These mappings have been characterized with the help of elementary operators of length 1. The above mentioned papers have motivated the investigation in the present paper.

AMS Subject Classification (1991): 47B47, 47B15

Received June 26, 2003, and in revised form September 24, 2003. (Registered under 5821/2009.)