I'm trying to prove the next:
If $S$ and $T$ are positive bounded self-adjoint linear operators on a complex Hilbert space $H$ and $S^{2} = T^{2},$ then $S = T.$
Because $S$ and $T$ are positive bounded self-adjoint linear operators, then $S^{2}$ and $T^{2}$ are too. So, there are unique positive square roots $A$ and $B$ such that $S^{2}=A^{2}$ and $T^{2}=B^{2},$ where $A=(S^{2})^{1/2}$ and $B=(T^{2})^{1/2}.$ Then, for uniqueness of positive roots, $A=B,$ i.e. $S=T.$
I feel uncomfortable with the last step, because, for example, $A=(S^{2})^{1/2}$ is only a symbol to denote such root, but does not mean that "algebraic operations" hold for this operator, isn'it?
I've tried to use the positive square roots of $T$ and $S$ with the above but I don't get any useful.
Any kind of help is thanked in advanced.