2009/01/07 by Djakov, Plamen, Mityagin, Boris
#34L40 #47B06 #47E05 #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.0901.0856
The Dirac operators Ly = i 1 amp; 0 0 amp; -1 (dy)/(dx) + v(x) y, y = y1 y2, x∈[0,π], with L2-potentials v(x) = 0 amp; P(x) Q(x) amp; 0, P,Q ∈ L2 ([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-Lbc)-1 dz, Pn = (1)/(2πi) ∫_|z-n|= 1/4 (z-Lbc)-1 dz are well--defined for |n| ≥ N if N is sufficiently large. It is proved that ∑|n| gt; N ‖Pn - Pn0‖2 lt; ∞, where Pn0, n ∈ ℤ, are the Riesz projections of the free operator. Then, by the Bari--Markus criterion, the spectral Riesz decompositions f = SN f + ∑|n| gt;N Pn f, ∀ f ∈ L2; converge unconditionally in L2.