Bari-Markus property for Riesz projections of Hill operators with singular potentials

Abstract

The Hill operators Ly = -y '' + v(x)y, x is an element of [0, pi], with H-1 periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large N, the Riesz projections P-n = 1/2 pi i integral(Cn) (z-L)(-1)dz, C-n = {z: |z-n(2)| = n} are well defined. It is proved that Sigma(n>N) parallel to P-n - P-n(0)parallel to(2) < infinity, where P-n(0) are the Riesz projections of the free operator.