1. square root lemma

Hello,

in the book functional analysis by reed, there is a lemma which says:
If $\displaystyle A \in L(H)$ is a bounded linear operator on a Hilbert Space, and$\displaystyle A\ge0$. Then there is a unique $\displaystyle $$B \in L(H)$$ B\ge0$ s.t. $\displaystyle B^2 = A.$

I have a question about the proof. The author argues:
"It is sufficient to consider the case where$\displaystyle \|A\| \ge 1$."

Why is it sufficient to proof just this case?

I couldn't find the right argument.

Regards

2. Re: square root lemma

If the result has been shown for the operators which has a norm $\displaystyle \geq 1$, let $\displaystyle T\in\mathcal{B}(H)$, $\displaystyle T\geq 0$. If $\displaystyle T=0$ take $\displaystyle B=0$ and if $\displaystyle \lVert T\rVert \neq 0$ then $\displaystyle S:=\frac{T}{\lVert T\rVert}\in\mathcal{B}(H)$ and $\displaystyle S\geq 0$. We can find an unique $\displaystyle B\geq 0$ such that $\displaystyle B^2=S$ i.e. $\displaystyle \lVert T\rVert^2B^2 =T$. We take $\displaystyle B' := \lVert T\rVert B$.

3. Re: square root lemma

Thank you very much!