Prove that every positive semidefinite operator $T\in B(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space, has a unique positive-semidefinite $n$th root for every $n\in\Bbb{N}$.
My idea was to use the Spectral theorem - I know that since $T$ is positive semidefinite, it's self adjoint and its spectrum $\sigma(T)\subseteq[0,\infty)$. Since it's self adjoint, there exists a projection - valued measure $P_T$ s.t $$T=\int_{\sigma(T)} tdP_T$$ I want to show (or atleast I suspect it to be true) that $$\sqrt[n]{T}=\int_{\sigma(T)} \sqrt[n]{t}dP_T$$ Which is well defined since $\sigma(T)\subseteq [0,\infty)$. Anyway, I didn't know exactly where to start (or even if this is true). Any help would be appreciated.