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

The Dirac operators Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi], with $L^2$-potentials v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,\pi]), considered on $[0,\pi ]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$S_N=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }z\mathrm{\mid }=N-1\mathrm{/}2}(z-L_{bc}{)}^{-1}dz,P_n=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }z-n\mathrm{\mid }=1\mathrm{/}4}(z-L_{bc}{)}^{-1}dz$$ are well–defined for $\mathrm{\mid }n\mathrm{\mid }\ge N$ if $N$ is sufficiently large. It is proved that $$\sum _{\mathrm{\mid }n\mathrm{\mid }>N}\mathrm{\parallel }{P}_n-P_n^0{\mathrm{\parallel }}^2<\mathrm{\infty },$$ where $P_n^0,n\in \mathbb{Z},$ are the Riesz projections of the free operator. Then, by the Bari–Markus criterion, the spectral Riesz decompositions $$f=S_{N}f+\sum _{\mathrm{\mid }n\mathrm{\mid }>N}{P}_{n}f,\mathrm{\forall }f\in L^2;$$ converge unconditionally in $L^2\mathrm{.}$