Index of an operator of separable Hilbert space

Question

I am trying to prove if H is a separable Hilbert space then the operator T in B(H) has the form U|T| (where U is a unitary operator and $|T|=(T^* T)^1/2$ ) iff index T=0. I was thinking to start with the polar decomposition of T i.e $T=U|T|$ but how should i proceed to index T=0.

Answer 1

For more see: Construction, Example, Overview

Hilbert Decomposition: $$X=\overline{\mathcal{R}|T|}\oplus\overline{\mathcal{R}|T|}^\perp\quad Y=\overline{\mathcal{R}T}\oplus\overline{\mathcal{R}T}^\perp$$

Square Root Lemma: $$T\in B(X,Y):\quad\left\|Tx\right\|=\left\||T|x\right\|$$ Unitary Operator: $$U:\overline{\mathcal{R}|T|}\leftrightarrow\overline{\mathcal{R}T}:\quad U|T|\varphi:=T\varphi$$

Range Relation: $$\overline{\mathcal{R}|T|}^\perp=\mathcal{R}|T|^\perp=\mathcal{N}|T|=\mathcal{N}T$$

Dimension Criterion: $$T=U|T|\iff\dim\mathcal{N}T=\dim\mathcal{R}T^\perp$$