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Let $T\in B(H)$ be a bounded operator on a Hilbert space. Then there exists a partial isometry $v$ such that $x=v|x|$ where $|x|=(x^*x)^{1/2}$ and this representation is essentially unique.

This decomposition is in analogy to the decomposition of a complex number $z$ as $z=|z| e^{i\varphi}$ where $\varphi$ is the argument of $z$. Now it is pretty obvious in which sense $|x|$ is the natural analogy of $|z|$ for an operator, however it is not clear to me in which sense a partial isometry is the pendant to a unit circle number.

Therefore I am interested in a motivation for this (despite that it works).

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    Note that non-zero complex numbers are invertible. If $x$ is invertible then $v$ is a unitary which corresponds to $z \bar z = 1$. If $x$ is not invertible one needs the weaker notion of partial isometries. – 2017-01-27
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    @AndréS.: exactly what I would have posted. You should have posted it as answer and not a comment, though. – 2017-01-27

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