My question is if we take $T$ in $B(H)$ with $\|T\|<1$.How can we show that the power series $$\sum_{n=0}^∞ \binom{1/2} n T^n$$ converges uniformly in $B(H)$ to an operator $(I + T)^{1/2}$ satisfying $((I + T)^{1/2})^2 = I + T$?
I also need to show that $(I + T)^{1/2} ≥ 0$ when T = T*. Here $H$ is a hilbert space. The hint is if we take $T=T^*$and $\|x\|=1$ then $((I + T)^{1/2}x|x)=1 - $$\sum_{n=0}^∞ \binom{1/2} n (T^nx|x)$.
Hint: Use Taylor expansion on $f(x)=\sqrt{1+x}$, $|x|<1$ (with error term).
In your hint for the self-adjoint case you should only sum from $n=1$ (and it should be a plus in front). Simply estimate the absolute value of each term and show that this sum does not exceed 1. Given the 1 outside their sum must be positive.