Example of Hilbert space operator that is not a product of unitary and positive

Question

If $A$ is a unital $C^{*}$-algebra, and $a\in A$ is invertible, then $a=u|a|$ where $u$ is unitary and $|a|=(a^{*}a)^{1/2}$ is positive. I am looking for an example of a bounded linear operator on some Hilbert space that is not a product of a unitary and a positive operator.

Answer 1

Consider the left shift $L$ on $\ell^2(\mathbb N)$, $$ L((a_1,a_2,a_3,\dots)) = (a_2,a_3,\dots) .$$ Suppose for contradiction that $L = U|A|$ where $U$ is unitary. Then $L^* L = |A|^2 = \text{diag}(0,1,1,\dots)$ which implies $\sigma(|A|^2) = \{0,1\}$. But $LL^* = U|A|^2 U^* = I$ which implies $\sigma(|A|^2) = \{1\}$. Hence contradiction.