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

摘要

The convergence of the series (3.1) is the analytic core of Bari-Markus Theorem (see [12], Ch. 6, Sect. 5.3, Theorem 5.2) which guarantees that the series ∑(P_nf) (|n|>N) converges unconditionally in L~2 for every f ∈ L~2. But in order to have the identity f = S_Nf + ∑(P_nf) (|n|>N), we need to check the "algebraic" hypotheses in Bari-Markus Theorem: (a) The system of projections {S_N; P_n, |n| > N} is complete, i.e., the linear span of the system of subspaces {E~*; E_n, |n| > N}, E~* = Ran S_N, E_n = Ran P_n, is dense in L~2(I). (b) The system of subspaces (5.2) is minimal, i.e., there is no vector in one of these subspaces that belongs to the closed linear span of all other subspaces. Condition (b) holds because the projections in (5.1) are continuous, commute and P_nS_N = 0, P_nP_m = 0 for m ≠ n, |m|, |n| > N. The system (5.1) is complete; this fact is well-known since the early 1950's (see details in [12, 15, 16]). More general statements are proven in [19] and [25], Theorems 6.1 and 6.4 or Proposition 7.1. Therefore, all hypotheses of Bari-Markus Theorem hold, and we have the following theorem.