Bari–Markus property for Riesz projections of 1D periodic Dirac operators

The Dirac operators Ly = i 1 0 0 −1 dy dx + v(x)y, y = y1 y2 , x ∈ [0, π], with L 2-potentials v(x) = 0 P (x) Q(x) 0 , P,Q ∈ L 2 ([0, π]), considered on [0, π] with periodic, antiperiodic or Dirichlet boundary conditions (bc), have discrete spectra, and the Riesz projections SN = 1 2πi |z|=N− 1 2 (z − L bc) −1 dz, Pn = 1 2πi |z−n|= 1 2 (z − L bc) −1 dz are well-defined for |n| ≥ N if N is sufficiently large. It is proved that |n|>N Pn − P 0 n 2 < ∞, where P 0 n , n ∈ Z, are the Riesz projections of the free operator. Then, by the Bari–Markus criterion, the spectral Riesz decompositions f = SN f + |n|>N Pnf, ∀f ∈ L 2 ; converge unconditionally in L 2 .