Q:

Square root of a bounded operator in Hilbert space

Consider the power series expansion
$$
   \sqrt{1-z} = 1+\sum\limits_{k=1}^\infty c_k z^k,
$$
converging absolutely in the ball $|z| \leq 1$. Let $H$ be a Hilbert space and $A \in \mathcal L(H)$ a bounded linear operator with $\|\mathrm{Id}-A\| \leq 1$. Then the series 
$$
   1 + \sum\limits_{k=1}^\infty c_k (I-A)^k
$$
converges in norm to some $B \in \mathcal L(H)$. How to show that $B^2 = A$?
Clearly, we have
$$
   \biggl| \bigl(1-\|I-A\|\bigr) - \bigl(1+\sum_{k=1}^N c_k\|I-A\|^k  \bigr)^2 \biggr| \to 0, \quad N \to +\infty,
$$
but I need to show somehow that
$$
  \biggl\| (I-(I-A)) - \bigl(I+\sum\limits_{k=1}^Nc_k(I-A)^k\bigr)^2 \biggr\| \to 0, \quad N \to + \infty.
$$
If it is necessary, $A = A^*$ and $A \geq 0$.

A:

We have the following identity
$$
  1 - z = 1+\sum\limits_{N=1}^\infty \bigl( \sum\nolimits_{i+j=N} c_i c_j \bigr) z^N
$$
near zero in $\mathbb C$, where $c_0 = 1$, which implies 
$$
   \sum\nolimits_{i+j=N} c_ic_j = \begin{cases}-1, & N = 1, \\
                                               0, & N \geq 2.
                                   \end{cases}
$$
Now we have
$$
   B^2 = I+\sum_{N=1}^\infty \bigl(\sum\nolimits_{i+j=N} c_i c_j\bigr) (I-A)^N = I-I+A=A.
$$

