A completeness problem related to the Riemann hypothesis

It is proven that the set of eigenvectors and generalized eigenvectors associated to the non-zero eigenvalues of the Hilbert-Schmidt (non nuclear, non normal) integral operator on L 2(0, 1) [Aρ (α)f](θ) = ∫01 ρ (αθ/x)} f(x)dx where α∈]0,1[ and ρ(x) = x - [x] is the fractionary part function, is total in L 2(0, 1), but it is not part of a Markushevich basis in L 2(0, 1) and therefore, it is not a Schauder basis in L 2(0, 1). © 2005 Birkhäuser Verlag Basel/Switzerland.