Abstract. A classical result of Riesz [8] says that the product of two commuting positive operators is also positive. Wigner's generalization states that the selfadjoint product of atmost three positive operators is positive too. Bernau proved that the selfadjoint product of a positive operator and another operator with positive spectrum is automatically positive. We show that these statements remain true under further essential weakening of the assumptions. The selfadjoint product of a positive operator and another operator with spectrum without negative reals is also positive (Theorem 4). Consequently the selfadjoint product of two positive operators and an operator with closed numerical range without negative reals is automatically positive (Theorem 6).

AMS Subject Classification (1991): 47A05, 47A12

Received April 8, 1999, and in revised form July 5, 1999. (Registered under 2734/2009.)