Abstract. If $A$ and $B$ are compact operators on a Hilbert space, with singular values satisfying $s_j(A)=s_j(B)=s_j((A+B)/2)$, for all $j=1, 2,...$, then $A=B$. Two proofs, geometric and analytic, are given.

AMS Subject Classification (1991): 47B06, 47B07, 31A05

Received June 11, 1996. (Registered under 6132/2009.)