Square root of a positive operator

Summary

Every positive operator $A$ on a Hilbert space has exactly one positive operator $B$ with $B^2 = A$, written $A^{1/2}$ or $\sqrt{A}$. It extends the square root of a positive semidefinite matrix to infinite dimensions. The square root defines the absolute value $\abs{T} = (T^*T)^{1/2}$ of an operator, which enters the polar decomposition and the definition of singular values.