I'm looking for references wich treat the following properties:
Let H be a hilbert space. Let $T \in B(H)$ be an invertible operator. Suppose that $T= U |T|$ is the polar decomposition of T. Then we have:
-
$T^{-1} = U^*|T^{-1} |$ is the polar decomposition of $T^{-1}$.
-
$|T^{-1}|= {|T^*|}^{-1}.$
-
${({|T|}^{-1})}^\alpha = U^* {(|T^{-1}|)}^\alpha U$ , for each $\alpha >0$.
Could you please suggest me some references on this please.
Thank you !
Best Answer
In most textbooks, these would be exercises at best. Mostly, you need to notice that $$\tag1 |T^{-1}|^2=(T^{-1})^*T^{-1}=(T^*)^{-1}T^{-1}=(TT^*)^{-1}=|T^*|^{-2}, $$ which is 2.
From the fact that $T$ is invertible, it follows that $U$ is a unitary: $$ U^*U=(T|T|^{-1})^*T|T|^{-1}=|T|^{-1}T^*T|T|^{-1}=|T|^{-1}|T|^2|T|^{-1}=I, $$ and $U$ is surjective because $T$ is.
Now, from $T=U|T|$ we get $T^{-1}=|T|^{-1}U^*$; and then $$ |T^{-1}|^2=(|T|^{-1}U^*)^*|T|^{-1}U^*=U|T|^{-2}U^*=(U|T|^{-1}U^*)^2, $$ so $$\tag2|T^{-1}|=U|T|^{-1}U^*.$$ Then $$ T^{-1}=|T|^{-1}U^*=U^*|T^{-1}|UU^*=U^*|T^{-1}| $$ is the polar decomposition.
Finally, item 3 follows directly from $(2)$.