Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in \mathbb{R}$. I want to show $T^{\alpha}$ is positive.
I used continuous functional calculus $\left(\sigma (f(T))=f(\sigma (T)\right)$ to show. I took $f(z)=z^{\alpha}$ which is continuous for $z\geq0$.
Is my proof correct?