Some results and a conjecture on the degree of ill-posedness for integration operators with weights

In this paper, we are looking for answers to the question whether a non-compact linear operator with non-closed range applied to a compact linear operator mapping between Hilbert spaces can, in a specific situation, destroy the degree of ill-posedness determined by the singular value decay rate of the compact operator. We partially generalize a result of Vu Kim Tuan and Gorenflo (1994 Inverse Problems 10 949-55) concerning the non-changing degree of ill-posedness of linear operator equations with fractional integral operators in L-2(0, 1) when weight functions appear. For power functions m(t) = t(alpha) (alpha > -1), we prove the asymptotics sigma(n) (A) similar to integral(0)(1) m(t) dt/pi n for the singular values of the composite operator [Ax](s) = m(s) integral(0)(s) x(t) dt in L-2(0, 1). We conjecture this asymptotic behaviour also for exponential functions m(t) = exp(-1/t(alpha)) (alpha > 0) that play some role for the local degree of ill-posedness for a nonlinear inverse problem in option pricing in Hein and Hofmann (2003 Inverse Problems 19 1319-38).