Spectrality of Ordinary Differential Operators

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford’s spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein’s conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff’s solutions of the equation l(y) = λy, which was discovered earlier by the author.