For positive operators $A$ and $B$ with $A^6=B^6$ show that $A=B$ Since $A$ and $B$ are positive, I managed to show that $A^6$ and $B^6$ are positive.
Now, I can use the fact that there exists a unique square root of both of those and since they're equal, their roots must be equal, so $A^3=B^3$. But what now?
I'm guessing I need to show that $A^2=B^2$ somehow since then I can say 
$A^3=B^3\Rightarrow AA^2=BB^2 \Rightarrow A=B$
How can I do that?

More generally, let $f$ be any strictly increasing continuous function on $[0, \infty)$ and suppose $f(A) = f(B)$ (as defined by the continuous functional calculus) where $A$ and $B$ are positive operators.  $f$ has a continuous inverse function $g$ defined on $f([0,\infty))$, and since $g(f(t)) = t$ for $t \in [0,\infty)$ we have $A = g(f(A)) = g(f(B)) = B$.
