functional, compact operator I am working on a homework problem (Analysis now E3.3.7) and I have no idea on how to solve it. Can anyone give some thoughts? Many thanks.
Assume that Hilbert pace $H$ is separable and prove that an operator $T$ in $B(H)$
has the form $U|T|$ for some unitary operator $U$ with the $|T|=(T^*T)^{\frac{1}{2}}$ iff index$T = 0$.
I know if $T$ is invertible, then $U$ in polar decomposition is unitary. But how to prove $T$ is invertible if index$T = 0$. And I have no idea on the other direction.

Remaining spaces:
$$\overline{\mathcal{R}|A|}^\perp=\mathcal{N}|A|=\mathcal{N}A\quad\mathcal{N}A^*=\overline{\mathcal{R}A}^\perp$$
For equal dimensions:
$$\dim\mathcal{N}A=\dim\mathcal{N}A^*\implies UU^*=1$$
For more details: Polar Decomposition
