Related papers: Right limits and reflectionless measures for CMV m…

We study right limits of the Bergman Shift matrix. Our results have applications to ratio asymptotics, weak asymptotic measures, relative asymptotics, and zero counting measures of the orthogonal and orthonormal polynomials.

We consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from…

We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices. We proceed by proving several variational estimates on the…

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $\Sigma_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure…

We study spectrum of finite truncations of unbounded Jacobi matrices with periodically modulated entries. In particular, we show that under some hypotheses a sequence of properly normalized eigenvalue counting measures converge vaguely to…

This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the {\omega} limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full…

We propose some backward-forward martingale decompositions for functions of reversible Markov chains. These decompositions are used to prove the functional CLT for reversible Markov chains with asymptotically linear variance of partial…

We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic…

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application…

The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove…

We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of…

We extend the spectral method for proving limit theorems to random non-uniformly expanding dynamical systems. This yields the CLT and moderate deviations principles (MDP). We show that as the amount of non-uniformity decreases the CLT rates…

This paper studies the asymptotic spectral properties of a renormalized sample correlation matrix, including the limiting spectral distribution, the properties of largest eigenvalues, and the central limit theorem for linear spectral…

For full-line Jacobi matrices, Schr\"odinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of $m$-functions, is equivalent to a lack of reflection in the dynamics in the sense that…

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a…

Non-asymptotic theory of random matrices strives to investigate the spectral properties of random matrices, which are valid with high probability for matrices of a large fixed size. Results obtained in this framework find their applications…

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators…

We develop subordinacy theory for extended CMV matrices. That is, we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the…

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a…

We investigate the symmetries of so-called generalized extended CMV matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant…