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 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, Weinho

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.

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi], considered with $L^2$-potentials v(x) = 0 & P(x) Q(x) & 0 and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ the spectrum of the free operator $L_{bc}(0)$ is simple while the spectrum of $L_{bc}(v)$ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,\pi].$ Analogous results are obtained for regular but not strictly regular $bc.

Abel-summability of eigenfunction expansions of three-point boundary value problems

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