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

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Official URL: http://dx.doi.org/10.1002/mana.200910003

The Dirac operators Ly = i ((1)(0) (0)(-1))dy/dx + v(x)y, y = ((y1)(y2)), x is an element of[0, pi], with L-2-potentials v(x) = ((0)(Q(x)) (P(x))(0)), P, Q is an element of L-2([0, pi]), considered on [0, pi] with periodic, antiperiodic or Dinchlet boundary conditions (bc), have discrete spectra, and the Riesz projections, S-N = 1/2 pi iota integral(vertical bar z vertical bar=N - 1/2) (z - L-bc)(-1) dz. p(n) = 1/2 pi iota integral(vertical bar z-n vertical bar=1/2) (z - L-bc)(-1) dz are well-defined for vertical bar n vertical bar >= N if N is sufficiently large. It is proved that Sigma(vertical bar n vertical bar>N) parallel to P-n - P-n(0)parallel to(2) < infinity, where P-n(0), n is an element of Z, are the Riesz projections of the free operator. Then, by the Ban Markus criterion, the spectral Riesz decompositions f = SN + Sigma(vertical bar n vertical bar>N) P(n)f, for all f is an element of L-2 converge unconditionally in L-2. (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinhom