Related papers: The Spectral Scale and the k-Numerical Range

We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear…

The main object of this work is the top-dimensional Laplacian operator of a simplicial complex $K$. We study its spectral limiting behavior under a given non-trivial subdivision procedure $\text{div}$. It will be shown that in case…

It was recently shown that the point spectrum of the separated Coulomb-Dirac operator H_0(k) is the limit of the point spectrum of the Dirac operator with anomalous magnetic moment H_a(k) as the anomaly parameter tends to 0; this spectral…

Let $\mathcal{A}$ be a $C^*$-algebra of bounded uniformly continuous functions on $X=\mathbb{R}^d$ such that $\mathcal{A}$ is stable under translations and contains the continuous functions that have a limit at infinity. Denote…

Let $-\lambda_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrt\lambda_j t}$ is the trace of the solution operator of the wave equation on the disk. It is…

We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral…

Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of $L$-functions. While most ensembles studied have continuous spectral distribution, Burkhardt et al introduced the ensemble of…

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n \lambda_i^k$, where $\lambda_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known…

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}\in\mathbb{C}^{n\times n}$ be two normal matrices with spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. The celebrated Hoffman--Wielandt…

We study the Point/Tukey spectrum of a general directed set using PCF theoretic tools and uncover basic connections between the theories. In particular, we prove that if the supremum of the Tukey spectrum is singular, then its cofinality…

Given a circulant matrix $\mathrm{circ}(c,a,0,0,...,0,a)$, $a\ne 0$, of order~$n$, we ``border'' it from left and from above by constant column and row, respectively, and we set the left top entry to be $-nc$. This way we get a~particular…

For a separable complex Hilbert space $H$, we say that a bounded linear operator $T$ acting on $H$ is $C$-normal, where $C$ is a conjugation on $H$, if it satisfies $CT^*TC=TT^*$. For a normal operator, we give geometric conditions which…

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ \lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{…

For any n-by-n complex matrix T and any $1\leqslant k\leqslant n$, let $\Lambda_{k}(T)$ the set of all $\lambda\in \C$ such that $PTP=\lambda P$ for some rank-k orthogonal projection $P$ be its higher rank-k numerical range. It is shown…

It is shown that for a given infinite graph $G$ on countably many vertices, and a compact, infinite set of real numbers $\Lambda$ there is a real symmetric matrix $A$ whose graph is $G$ and its spectrum is $\Lambda$. Moreover, the set of…

Given a subspace $U\subset\mathbb{C}[x_1,\dots,x_n]_d$ we consider the closure of the image of the rational map $\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{\dim U-1}$ given by $U$. Its coordinate ring is isomorphic to $\bigoplus_{i\ge 0}…

We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is…

The restricted numerical range $W_R(A)$ of an operator $A$ acting on a $D$-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset $R$ of the of…

We study the relation between the spectrum of a self-adjoint operator and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the…