Polar Decomposition: Unitarity

Question

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).

Answer 1

Left Shift

Denote for shorthand: $$1_0(0,x_1,\ldots):=(0,x_1,\ldots)$$

For the modulus: $$L^*L=RL=1_0\implies|L|=1_0$$

For the argument: $$L=U|L|\implies U=1_0,1,\ldots$$

So it admits one.

Right Shift

For the modulus: $$R^*R=LR=1\implies|R|=1$$

For the argument: $$R=U|R|\implies U=R$$

So it admits none.

Reference

For much more details: Polar Decomposition