An Application of the Theorem of Rudin to the Toeplitz Operators on Odd Spheres.
Selfadjoint operator matrices with finite rows
A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.
A Weyl-Titchmarsh type formula for a discrete Schrödinger operator with Wigner-von Neumann potential
We consider a discrete Schrödinger operator 𝒥 with Wigner-von Neumann potential not belonging to l². We find the asymptotics of orthonormal polynomials associated to 𝒥. We prove a Weyl-Titchmarsh type formula, which relates the spectral density of 𝒥 to a coefficient in the asymptotics of the orthonormal polynomials.
Spectral analysis of unbounded Jacobi operators with oscillating entries
We describe the spectra of Jacobi operators J with some irregular entries. We divide ℝ into three “spectral regions” for J and using the subordinacy method and asymptotic methods based on some particular discrete versions of Levinson’s theorem we prove the absolute continuity in the first region and the pure pointness in the second. In the third region no information is given by the above methods, and we call it the “uncertainty region”. As an illustration, we introduce and analyse the OP family...
Unbounded Jacobi Matrices with Empty Absolutely Continuous Spectrum
Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.
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