Related papers: The spectrum of a random operator is a random set
This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of "Brown measure," which play…
The boundary operator is a linear operator that acts on a collection of high-dimensional binary points (simplices) and maps them to their boundaries. This boundary map is one of the key components in numerous applications, including…
In the Contextuality-by-Default theory random variables representing measurement outcomes are labeled contextually, i.e., not only by what they measure but also under what conditions (in what contexts) the measurements are made, including…
Of crucial importance to the development of quantum computing and information has been the construction of a quantum operations formalism that admits a description of quantum noise while simultaneously revealing the behavior of an open…
This note is a survey and collection of results, as well as presenting some original research. For Bessel sequences and frames, the analysis, synthesis and frame operators as well as the Gram matrix are well-known, bounded operators. We…
Many studies have been conducted on statistical convergence, and it remains an area of active research. Since its introduction, statistical convergence has found applications many fields. Nevertheless, there is a shortage of research…
We propose a theoretical framework to study the eigenvalue spectra of the controllability Gramian of systems with random state matrices, such as networked systems with a random graph structure. Using random matrix theory, we provide…
We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is…
Let $A\colon H\rightarrow H$ be a normal operator on an infinite-dimensional separable Hilbert space $H$ and let $S\subseteq H$ be a finite subset such that $\{A^nx\}_{n\geq 0,\,x\in S}$ can be rescaled to form a frame for $H$. That is,…
Random matrix theory (RMT) provides a framework to study the spectral fluctuations in physical systems. RMT is capable of making predictions for the fluctuations only after the removal of the secular properties of the spectrum. Spectral…
Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distributions of large self-adjoint random matrices from the generalized unitary ensembles. This paper gives sufficient conditions for an…
In this note we introduce the concept of the numerical range of a bounded linear operator with respect to a family of projections. We give a precise definition and elaborate on its connection to the classical numerical range as well as to…
Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is…
We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically…
Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and…
Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per…
In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
This work starts from definition of randomness, the results of algorithmic randomness are analyzed from the perspective of application. Then, the source and nature of randomness is explored, and the relationship between infinity and…
We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic…