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

Abstract

The Dirac operatorsconsidered on [0, π] with periodic, antiperiodic or Dirichlet boundary conditions (bc), have discrete spectra, and the Riesz projectionswhere P 0 n , n ∈ Z, are the Riesz projections of the free operator. Then, by the Bari-Markus criterion, the spectral Riesz decompositionsconverge unconditionally in L 2 .

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

“…Basically they are devoted to Riesz basis property of EAF for BVP with strictly regular (and just regular) BC for 2 × 2 Dirac systems. The most complete and detailed results in this direction have been obtained by P. Djakov and B. Mityagin [5][6][7][10][11][12][13]. In recent preprint [12] they proved equiconvergence and pointwise convergence of spectral decompositions of Dirac operators with regular BC.…”

“…The Riesz basis property of EAF for BVP with regular but non-strictly regular (including periodic, antiperiodic and other) BC for 2 × 2 Dirac systems have been investigated by P. Djakov and B. Mityagin [39,7,10]. Namely, in [39] and [7] they proved the Riesz basis property of subspaces (spectral projections) for 2 × 2 Dirac system with periodic and antiperiodic BC.…”

“…Namely, in [39] and [7] they proved the Riesz basis property of subspaces (spectral projections) for 2 × 2 Dirac system with periodic and antiperiodic BC. In the next publication [10] these authors extended their result to the case of arbitrary regular but not strictly regular BC.…”

“…In connection with Corollary 1.3 and Proposition 1.4 we mention the papers [49,50,21,39] and [6][7][8][9][10][11][12][13], that appeared during the last decade. Basically they are devoted to Riesz basis property of EAF for BVP with strictly regular (and just regular) BC for 2 × 2 Dirac systems.…”

Метод Подобных Операторов В Спектральном Анализе Несамосопряженного Оператора Дирака С Негладким Потенциалом

“…Авторы выражают благодарность рецензенту, обратившему наше внимание на статью [14], где получен результат следствия 3. Из этого результата можно с помощью теоремы Бари-Маркуса вывести утверждение теоремы 8.…”