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 \textit{almost orthogonality} of Birkhoff's solutions of the equation $l(y)=\lambda y$, which was discovered earlier by the author.Comment: AmsLaTeX, 26 pages, added section on dissipative operators and reference

Spectral properties of truncated Toeplitz operators by equivalence after extension

We study truncated Toeplitz operators in model spaces View the MathML source for 1<p<∞, with essentially bounded symbols in a class including the algebra View the MathML source, as well as sums of analytic and anti-analytic functions satisfying a θ -separation condition, using their equivalence after extension to Toeplitz operators with 2×2 matrix symbols. We establish Fredholmness and invertibility criteria for truncated Toeplitz operators with θ -separated symbols and, in particular, we identify a class of operators for which semi-Fredholmness is equivalent to invertibility. For symbols in View the MathML source, we extend to all p∈(1,∞) the spectral mapping theorem for the essential spectrum. Stronger results are obtained in the case of operators with rational symbols, or if the underlying model space is finite-dimensional

Interpolation by vector-valued analytic functions, with applications to controllability

In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the controllability properties of linear semigroup systems with a Riesz basis of eigenvectors