 Singular system of a compact operator

For a compact operator $A:X\to Y$ between Hilbert spaces, a singular system consists of positive singular values and orthonormal families withThe $v_j$ form a basis of $(\ker A{)}^{\perp }$, and the $u_j$ form a basis of the closure of its operator range. The family is finite when $A$ has finite rank. For infinite rank, the singular values can be ordered with ${\sigma }_j\downarrow 0$.

The expansion $Ax={\sum }_j{\sigma }_j⟨x,v_j⟩u_j$ separates channels with different data sensitivity. Inverting a channel divides by ${\sigma }_j$, explaining why an infinite-rank compact operator generally has an unbounded inverse on its range. Spectral regularization methods replace those inverse coefficients by bounded filters.

For the diagonal operator on sequence space $A(x_1,x_2,\dots )=(x_1,x_2/2,x_3/3,\dots )$ on L2 space, the standard coordinate vectors give ${\sigma }_j=1/j$. A small error in a late output coordinate can correspond to a much larger change in the unknown. No infinite sequence tending to zero is required for a finite-rank operator.