Related papers: Spectral Sets: Numerical Range and Beyond
In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a…
In this paper we prove a conjecture stated by the first two authors establishing the closure of the numerical range of a certain class of $n+1$-periodic tridiagonal operators as the convex hull of the numerical ranges of two tridiagonal…
We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give…
They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving w_N. We also study particular cases with a fixed N(.), for…
Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave…
We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$…
The $k$-rank numerical range $\Lambda_{k}(A)$ is expressed via an intersection of a countable family of numerical ranges $\{F(M^{*}_{\nu}AM_{\nu})\}_{\nu\in\mathbb{N}}$ with respect to $n\times (n-k+1)$ isometries $M_{\nu}$. This…
Nonlinear approximation from regular piecewise polynomials (splines) of degree $<k$ supported on rings in $\R^2$ is studied. By definition a ring is a set in $\R^2$ obtained by subtracting a compact convex set with polygonal boundary from…
Here we show that for $k\in \mathbb N,$ the closure of the $k$-rank numerical range of a contraction $A$ acting on an infinite-dimensional Hilbert space $\mathcal{H}$ is the intersection of the closure of the $k$-rank numerical ranges of…
In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of…
For a given set of $n\times n$ matrices $\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\mathcal F$, denoted by $W_C({\mathcal F})$. We obtain basic algebraic and topological properties of…
The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$…
This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.
We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential…
The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the…
For a positive integer $k$, the rank-$k$ numerical range $\Lambda_k(A)$ of an operator $A$ acting on a Hilbert space $\cH$ of dimension at least $k$ is the set of scalars $\lambda$ such that $PAP = \lambda P$ for some rank $k$ orthogonal…
The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \geq 0$ such that $A$ is contained in a translate of $\lambda K$. Each choice of $K$ gives a…
In this short note we first extend the validity of the spectral radius formula obtained in \cite{ag} to Fourier--Stieltjes algebras. The second part is devoted to showing that for the measure algebra on any locally compact non-discrete…