1. ## square root lemma

Hello,

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

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

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 $\geq 1$, let $T\in\mathcal{B}(H)$, $T\geq 0$. If $T=0$ take $B=0$ and if $\lVert T\rVert \neq 0$ then $S:=\frac{T}{\lVert T\rVert}\in\mathcal{B}(H)$ and $S\geq 0$. We can find an unique $B\geq 0$ such that $B^2=S$ i.e. $\lVert T\rVert^2B^2 =T$. We take $B' := \lVert T\rVert B$.

3. ## Re: square root lemma

Thank you very much!