I've been reading bounded operators in Hilbert/Banach spaces and have come across the notation $\left| A \right| = \sqrt{A^{\ast}A}$ for some operator $A$. I haven't really found a good explanation about what this means for an operator however.
In particular what this would mean for a self adjoint operator $A$. I don't think we would simply have that $\left| A \right| = \sqrt{A^{\ast}A} = \sqrt{A^2} = A.$
If $A$ is a bounded linear operator on a Hilbert space $H$ , then $B:=A^{\ast}A$ is positive (this means $(Bx,x) \ge 0$ for all $ x \in H$).
One can show: there is exacrtly one positiv bounded linear operator $Q$ such that $Q^2=B$.
Then define: $|A|:=Q$
You are right, that we not always have
(*) $\left| A \right| = \sqrt{A^{\ast}A} = \sqrt{A^2} = A$ if $A$ is selfadiont.
(*) is only valid , if $A$ is positive