Bari-Markus property for Riesz projections of 1D periodic Dirac operators
| dc.creator | Djakov, Plamen | |
| dc.creator | Mityagin, Boris | |
| dc.date | 2009-01-07 | |
| dc.date.accessioned | 2026-07-07T12:27:11Z | |
| dc.date.available | 2026-07-07T12:27:11Z | |
| dc.description | The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,π]), $$ considered on $[0,π]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$ S_N = \frac{1}{2πi} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2πi} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$ 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 $L^2.$ | |
| dc.identifier | https://arxiv.org/abs/0901.0856 | |
| dc.identifier | http://arxiv.org/abs/0901.0856 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/215121 | |
| dc.subject | Spectral Theory | |
| dc.subject | 34L40, 47B06, 47E05 | |
| dc.title | Bari-Markus property for Riesz projections of 1D periodic Dirac operators | |
| dc.type | text |