We prove new spectral enclosures for the non-real spectrum of a class of $2times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. One of our main results resembles Gershgorins circle theorem. The enclosures are applied to $J$-frame operators.
We consider a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, we state localisation theorems for the pair-eigenvalues and discuss relations to a class of non-self-adjoint spectral problems.
We construct a functional model (direct integral expansion) and study the spectra of certain periodic block-operator Jacobi matrices, in particular, of general 2D partial difference operators of the second order. We obtain the upper bound, optimal in a sense, for the Lebesgue measure of their spectra. The examples of the operators for which there are several gaps in the spectrum are given.
We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of orthogonal polynomials associated with the matrix.
We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic (double root) situation. For the model with non-smooth matrix entries we obtain the asymptotics of generalized eigenvectors and analyze the spectrum. In addition, we reformulate a very helpful theorem from a paper of Janas and Moszynski in its full generality in order to serve the needs of our method.