$T$ and $S$ are two linear self-adjoint and positive and bounded operators defined on complex Hilbert space $H$. show that if we have $T^2 = S^2$ then $T=S$.
I know that $T^2$ and $S^2$ should have the same characteristic function and eigenvectors. So should we conclude that the same thing for eigenvectors of $T$ and $S$? And then what can we do?
Every positive element in a $C^*$-algebra has a unique positive square root. Your assumption means that the positive operator $T^2$ has positive square roots $S$ and $T$. Hence, $S = T$.