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 $$S_N = \frac{1}{2\pi i} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2\pi i} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz$$ are well--defined for $|n| \geq N$ if $N$ is sufficiently large. It is proved that $$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty,$$ 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 $$f = S_N f + \sum_{|n| >N} P_n f, \quad \forall f \in L^2;$$ converge unconditionally in $L^2.$