Nonlinear ill-posed equations: Singular value decomposition and the Picard criterion (Q1084836)

A representation of a compact (nonlinear) operator F:A\(\to Y\) in the form \(F(x)=\sum \phi_ n(x)y_ n\) is discussed and the existence of sequences \((x_ n)\subset A\), \((y_ n)\subset Y\) and \((\lambda_ n)\subset {\mathbb{R}}\), such that \(F(x_ n)=\lambda_ ny_ n\) which is the singular value decomposition, if F is additionally linear. Furthermore it is shown that \[ Range F\subset \{y=\sum^{\infty}_{k-1}\eta_ ky_ k: \forall n\in {\mathbb{N}}\sum^{\infty}_{k=n}| \eta_ k|^ 2\leq \lambda^ 2_ n\} \] which is similar to the Picard criterion for compact linear operators.