相关论文: Concavity of Eigenvalue Sums and the Spectral Shif…

Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any $\alpha\in\RR\backslash{0}$. We will here study the problem of finding $(\lambda,v)\in\RR\times…

For a given $\delta$, $0<\delta<1$, a Blaschke sequence $\sigma=\{\lambda_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $\delta<\delta_f=\inf_{\lambda\in\sigma}|f(\lambda)|\le\|f\|_\infty\le1$ is invertible in…

We study the semigroup of the symmetric $\alpha$-stable process in bounded domains in $\R^2$. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This…

We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.

A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real…

The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note…

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and…

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real…

Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. The fundamental question arises: How does the singular value spectrum of the…

We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our…

When can one change the diagonal of a matrix without changing its spectrum? We completely answer this question over an algebraically closed field of characteristic zero or larger than the size of the matrix: An $n \times n$ matrix $A$…

Given a right eigenvector $x$ and a left eigenvector $y$ associated with the same eigenvalue of a matrix $A$, there is a Hermitian positive definite matrix $H$ for which $y=Hx$. The matrix $H$ defines an inner product and consequently also…

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient…

A Hankel operator $\Gamma$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $\Gamma$ is positive semi-definite if and only if $h$ is…

In \cite{CGPWW2021}, it was conjectured that a particular shifted sum of even divisor sums vanishes, and in \cite{SDK}, a formal argument was given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier…

We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities.

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.

The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for…

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on…