Singular system of a compact operator

A singular system $({\sigma }_j,u_j,v_j)$ for a compact operator $A:X\to Y$ has positive singular values ${\sigma }_j\to 0$ and orthonormal vectors satisfying $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$. It diagonalizes spectral regularization methods.