$T=AU \iff T $ is a normal operator on Hilbert space

adjoint-operatorsfunctional-analysisoperator-theory

This is Exercise 16.(c) from Conway's Functional Analysis book.

Suppose $H$ is a Hilbert space and $T$ is a compact operator on $H$. Assuming the result that $\exists A$ positive operator and $U$ a bounded operator such that $T=UA$

(NOTE: here $A= \sqrt {T^* T} $ and $U$ is the operator defined as in the construction of the polar decomposition i.e. $\|Uh\|=\|h\|$ when $h \perp \ker(T)$ and $Uh=0$ when $h \in\ker(T)$ i.e. $U(Ah)=Th$ when $h \perp \ker(A)=\bar{Ran(A)}$ and $Uh=0$ when $h\in\ker(A)$ ) ,

I have to prove :

$T=AU \iff T $ is normal

I tried to compute products and tried to show the one direction that ($T=AU \implies T $ is normal) , but since $U$ is just assumed to be bounded, not unitary, got stuck!

EDIT :

For $T$ Normal ,assuming $U$ to be unitary ( we can do that provided $T$ is normal), have been able to show that $AU=UA$ .

But for the other direction, ($T=AU=UA \implies T$ normal), got, $T^*T=AU^*UA$ and $TT^* = AU^*UA$ . So if U is at least normal, we are done! (as mentioned in an answer) . But , we have to be careful, because by construction of $U$ and $A$, it is not clear to me how is $U$ normal. ( since we are trying to show that $T$ is normal, we cannot assume beforehand that $U$ is unitary, like the other direction)

Please help!

Best Answer

I read through the exercise in Conway's book
(Second edition from 1990, Chapter II, ยง7, Exercise 16, Part c, it's on page 59),
and it deals with the polar decomposition of $\,T=U\!A\,$ for some bounded operator $T$ on $H$. The bounded operators $A$ and $U$ are uniquely determined by $T$ when asked to satisfy

  • $A=\sqrt{T^*T}=|T| $
  • $U$ is a partial isometry with $\,\ker U=\ker T\,$ and $\,\operatorname{im} U =\overline{\,\operatorname{im}T}\,$.

Note that $\,U^*\!\!\:U\,$ is the projector onto $\,\overline{\operatorname{im}|T|}\,$ which implies $\,A=U^*\!\!\:U\!A=U^*T$.
Multiplying with $U$ from the left yields $\,UU^*T=T$.

Consequences for the polar decomposition of $\,T^*$ $$=(U\!A)^*=AU^*= U^*\big(U\!AU^*\big)$$ Thus $\,U|T|U^*=\,|T^*|\,$ and $\,T^*=U^*|T^*|\,$ is the (unique) polar decomposition of $\,T^*$.

In the sequel $T$ is not required to be compact. I would guess that mentioning that property is due to the early placement of the exercise in the book in which Conway proceeds from the special to the more general.

It is shown that $\:T=AU\iff T$ is normal.

"$\,\Longrightarrow\,$"
If $T=U\!A=AU\,$ then $T^*=AU^*=U^*\!A$, so the assumption tells us that $A$ also commutes with $U^*$. Hence $$\,T^*= U^*\big(U\!AU^*\big)= U^*UU^*\!A = U^*\!A$$ is the polar decomposition of $\,T^*$. We have $\,UT^*=A\,$ by the above "Note that ..." applied to $\,T^*$, and conclude that $T\,T^*=A\,UT^*=A^2=T^*T$.

"$\,\Longleftarrow\,$"
$\,T=U\!A\,$ yields $\,T^*=AU^*$. Hence $\,UA^2U^* =T\,T^*=T^*T= A^2\,$ by normality. Taking the positive square-root (which is unique) gives $\,U\!AU^* = A$, being equivalent to $\,UA = A\,U$ by the properties of $U$.

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