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

Abstract

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 $[0,\pi ]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$S_N=\frac{1}{2\pi i}{\int }_{|z|=N-1/2}(z-L_{bc}{)}^{-1}dz,P_n=\frac{1}{2\pi i}{\int }_{|z-n|=1/4}(z-L_{bc}{)}^{-1}dz$$ are well--defined for $|n|\ge N$ if $N$ is sufficiently large. It is proved that $$\sum _{|n|>N}\Vert P_n-P_n^0{\Vert }^2<\infty ,$$ 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 $$f=S_{N}f+\sum _{|n|>N}{P}_{n}f,\forall f\in L^2;$$ converge unconditionally in $L^2.$

References in corpus (1)

• Deviations of Riesz projections of Hill operators with singular potentials