Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices

It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial, then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of $A$ is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure

The polyanalytic Ginibre ensembles

We consider a polyanalytic generalization of the Ginibre ensemble. This models allowing higher Landau levels (the Ginibre ensemble corresponds to the lowest Landau level). We study the local behavior of this point process under blow-ups.Comment: 31 page

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

We survey the current status of universality limits for $m$-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider $\beta$ ensembles for $\beta \neq 2$, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems

Localization and Toeplitz Operators on Polyanalytic Fock Spaces

The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $\psi$ and symbol ${\bf a}(x,\omega)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $\Psi$ and symbol $\sigma(z,\bar{z})$ coincides with a {\it Toeplitz} operator with symbol $D\sigma(z,\bar{z})$ for some polynomial differential operator $D$. Using the Berezin quantization approach, we will extend the proof for polyanalytic Fock spaces. While the generation is almost mimetic for two-windowed localization operators, the Gabor analysis framework for vector-valued windows will provide a meaningful generalization of this conjecture for {\it true polyanalytic} Fock spaces and moreover for polyanalytic Fock spaces. Further extensions of this conjecture to certain classes of Gel'fand-Shilov spaces will also be considered {\it a-posteriori}.Comment: 23 page