Let $A, B$ be self-adjoint positive operators, $C, D$ be orthogonal operators on an Euclidean space, and $AC = DB$.
Prove that $C=D$.
I know many properties of self-adjoint and orthogonal operators and tried to apply them but with no success. Could you please help me?
UPD: From $AC = DB$, since $C$ is orthogonal, we have $A=DBC^*$. Hence, $A^*=CB^*D^*$, and since $A$ and $B$ are self-adjoint, $A=CBD^*$. Multiplying these two decompositions of $A$ we get $A^2=DB^2D^*=CB^2C^*$.
Can we now say that $C=D$?
I suppose that by "positive operators", you mean strictly positive (i.e. positive definite) linear operators. Otherwise the statement is false, such as when $A=B=\operatorname{diag}(1,0),C=I$ and $D= \operatorname{diag}(1,-1)$.
Since $A^2=(AC)(AC)^T=(DB)(DB)^T=DB^2D^T$ and positive operators have unique positive square roots, we have $A=DBD^T$ and hence $AD=DB$. Thus $AC\,(=DB)=AD$ and in turn, $C=D$.