Intuition for absolute value of a bounded operator in the context of polar decomposition Let $T$ be a bounded operator on Hilbert space. The functional calculus for bounded symmetric operators defines a positive symmetric $|T|=\sqrt{T^\ast T}$.
Different operators can have the same absolute value operator. For instance, let $T$ be the shift operator on $\ell^2(\mathbb N)$ given by padding by zero. Its adjoint $T^\ast $ omits the first coordinate. We have $T^\ast T=I$ so $|T|=|I|=I$.
Intuitively, what kind of information does $|T|$ see that its notation might suggest?
 A: The terminology of "polar form" deliberately parallels that of complex numbers. We can take our inspiration from $\Bbb{C}$ as well.
Think about the polar form of a complex number $re^{i\theta}$. Instead of thinking about this complex number as a number itself, think of it as acting on $\Bbb{C}$ by multiplication. What happens when we multiply a complex number $z$ by $re^{i\theta}$?
Well, given $r$ is a positive real number, multiplication by $r$ simply shrinks or stretches $z$. On the other hand, the $e^{i\theta}$ factor will rotate $z$, counter-clockwise, by $\theta$, without changing the length of $z$. In that way, multiplication in $\Bbb{C}$ can always be thought of as a composition of two types of operation: stretching/shrinking, and isometric rotation.
Now, obviously on a general Hilbert space, operators are a little bit more complex: there are more than just compositions of rotations and scaling functions. However, if we are willing to expand on the idea of "scaling" and "rotation", we can indeed decompose operators in a similar way to complex numbers.
Instead of a rotation, we replace with the notion of an isometry. This is more broad than just a rotation; this also includes reflections, as well as various compositions of rotations that don't turn out to be rotations themselves. In either case, this is the part of the operator that changes directions of things, without changing their length.
Instead of a real scaling operation, we replace with the notion of a positive semi-definite operator. In finite-dimensions, such operators are diagonalisble. This means that we can decompose the space into eigenspaces, which are all scaled at various rates (according to the eigenvalues). All of these eigenvalues are non-negative reals, so like $r$ in the polar form $re^{i\theta}$, orientation is preserved, and all the eigenspaces are scaled positively.
In some ways, it's like various scaling factors applied simultaneously to various axes in the space. Even though eigenvectors are not a given in infinite dimensions, I believe it is still helpful to think of positive semi-definite operators in this kind of light.
So, what does $|T|$ encode? It's a bit of a tricky question to answer specifically, but it tells you a little bit about how vectors in the space stretch or shrink, before they're rotated away in various directions.
