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8 votes
2 answers
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Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

It is known that an entire function that is nowhere zero must be the exponential of another entire function. Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
Kanghun Kim's user avatar
7 votes
1 answer
2k views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
cs89's user avatar
  • 981
7 votes
0 answers
123 views

Steklov eigenvalue for circle valued functions

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization: $$\sigma_1(M,g)...
Eduardo Longa's user avatar
6 votes
1 answer
1k views

Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections $$P_{\{\lambda_1,...\lambda_n\}}=\frac{...
Frank's user avatar
  • 241
6 votes
0 answers
201 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
Manuel Cañizares's user avatar
6 votes
0 answers
188 views

Measurability of eigenvalues-eigenvectors of a positive compact operator

Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$. By the spectral theorem, given $a \in A$, there are scalars $...
user127022's user avatar
6 votes
0 answers
107 views

Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
Sascha's user avatar
  • 536
6 votes
0 answers
320 views

Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
Chilote's user avatar
  • 596
6 votes
0 answers
137 views

Spectrum of perturbed differential operators

I am looking for a reference that could help me with the following two questions: Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
Dominick's user avatar
5 votes
2 answers
1k views

Compact operator without eigenvalues?

Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$ $$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator! Then, let $l$ be the left-shift and $r$ ...
Landauer's user avatar
  • 173
5 votes
0 answers
250 views

Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb R}...
Gandalf Lechner's user avatar
4 votes
2 answers
4k views

Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. What is the eigenfunction of a multivariate Gaussian kernel: \begin{...
Amirreza Shaban's user avatar
4 votes
1 answer
728 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
4 votes
1 answer
274 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
4 votes
1 answer
120 views

Diagonalization of a specific Dirac operator

A few hours ago, a question was posed asking for the eigenvalues and eigenvectors of the Dirac operator $$ H=\begin{pmatrix} x & 0 & -i\partial_{x} & \bar{z} \\ 0 & x & z & i\...
Michael Engelhardt's user avatar
3 votes
1 answer
646 views

Eigenfunctions and eigenvalues of an operator defined by a certain integral

Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by $ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
hkju's user avatar
  • 245
3 votes
1 answer
255 views

Sums over products over short paths in an expander graph

Let $\Gamma=(V,E)$ be an undirected graph of degree $d$. (Say $d$ is a large constant and the number of vertices $n=|V|$ is much larger.) Let $W_0$ be the space of functions $f:V\to \mathbb{C}$ with ...
H A Helfgott's user avatar
  • 20.2k
3 votes
1 answer
264 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
Jan Rozendaal's user avatar
3 votes
0 answers
283 views

Extending Ky Fan's eigenvalues inequality to kernel operators

--Migrating from MSE since it might fit better here-- Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as: $$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
user43389's user avatar
  • 255
3 votes
0 answers
201 views

Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
dohmatob's user avatar
  • 6,853
3 votes
0 answers
52 views

Integral estimate in the Levitan's paper "On expansion in eigenfunctions of the Laplace operator"

Let $D \subset R^m$ be a domain in a $m$-dimensional Euclidean space, $P \in intD$, and $t > 0$ so small that the sphere of radius $t$ centered at the point $P$ sits in $intD$. Let $\phi : D \...
user109413's user avatar
3 votes
0 answers
190 views

Error term in the Euclidean Weyl law

Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
Dario's user avatar
  • 381
2 votes
1 answer
228 views

Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
Elio Li's user avatar
  • 809
2 votes
1 answer
756 views

General strategy for studying the decay of eigenvalues of kernel integral operators

Disclaimer. Please, be patient, I'm here to learn functional analysis... Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
223 views

Sobolev regularity via Laplace spectrum

Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
76 views

The eigenvalues of the product $WD$ for some particular matrices

Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0 Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is $$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
ABB's user avatar
  • 4,058
2 votes
1 answer
134 views

Common eigenvalues for two Sturm-Liouville problem

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form $$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
Gustave's user avatar
  • 617
2 votes
1 answer
346 views

Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
user avatar
2 votes
1 answer
172 views

Maximizing the first Neumann eigenvalue on disks

Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that $$\mu_1(g) \operatorname{...
Eduardo Longa's user avatar
2 votes
1 answer
135 views

Pseudo-polynomial potentials for Schrödinger operators

Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$. Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining ...
M. Veruete's user avatar
2 votes
0 answers
143 views

Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
builtdifferential's user avatar
2 votes
0 answers
43 views

Selecting some linearly independent columns of a particular matrix

Let us consider the matrix $C=A_1+A_2$ where : $A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$ $A_2$ is the the $n$ by $n$ block ...
ABB's user avatar
  • 4,058
2 votes
0 answers
81 views

Perturbed Gram matrix

Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix $$\sum_{t=1}^T(x_t ...
rostader's user avatar
  • 215
2 votes
0 answers
114 views

Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?

Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
150 views

Limit circle/point of an ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
xiaohuamao's user avatar
2 votes
0 answers
152 views

Relationship between eigenvalues of compact operators $A$ and $(A+A^*)/2$

A result from 'Topics in Matrix Analysis' by Horn & Johnson (3.3.33) is the following: For $A\in \mathbb{M}_n$, $\sum_{i=1}^k Re \lambda_i(A) \leq \sum_{i=1}^k Re \lambda_i \big(\frac{A+A^*}{2}\...
Shiba Biswal's user avatar
2 votes
0 answers
246 views

Decay rate of least eigenvalue of Gram matrices

Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$: $$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$ In ...
user39756's user avatar
  • 141
2 votes
0 answers
463 views

Conditions for continuity of non-simple eigenvectors

Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
billbob's user avatar
  • 37
2 votes
0 answers
259 views

Common eigenvector

I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem: let $V$ be an infinite-dimensional locally convex (but not normed!) ...
Alex M.'s user avatar
  • 5,407
1 vote
1 answer
119 views

Eigenvalues under linear transformation

Let $X$ and $Y$ be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small $\varepsilon>0$ such that, for any eigenvalue $\...
legon's user avatar
  • 31
1 vote
1 answer
195 views

Eigenvalues of operator

In the question here the author asks for the eigenvalues of an operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ Here I would like to ask if one can extend ...
Kung Yao's user avatar
  • 192
1 vote
1 answer
150 views

eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
user92646's user avatar
  • 617
1 vote
0 answers
125 views

Transforming nilpotency into diagonalizability [closed]

We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$. We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows: $Te_1=0$ and $...
ABB's user avatar
  • 4,058
1 vote
0 answers
34 views

Minimum eigenvalue of normal matrix with polynomial basis

For each $n\in \mathbb{N}\cup\{0\}$, let $x_n(t)=\frac{t^n}{n!}$ for all $t\in [0,1]$. As the functions $X_N=(x_0 ,\ldots, x_N)$ are linearly independent, the matrix $ \int_0^1 X_N(t)^\top X_N(t)\,\...
John's user avatar
  • 503
1 vote
0 answers
122 views

eigenvalues of integral operator with centered kernel

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that there is a reproducing kernel Hilbert Space $\...
Kcafe's user avatar
  • 519
1 vote
0 answers
130 views

Results on the eigenspace of weighted elliptic eigenvalue problems

I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$ $$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$ where the weights $a>0$ and $w\in L^{\infty}$ (and ...
Student's user avatar
  • 537
1 vote
0 answers
139 views

Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$

Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by $$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$ that is $$\mathcal{L}(...
Guilherme's user avatar
  • 205
1 vote
0 answers
181 views

Eigenvalues of product of operators

Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...
Kcafe's user avatar
  • 519
1 vote
0 answers
259 views

An estimate for the solution of an elliptic PDE depending on a parameter

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$. We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator $$ -\Delta:\ H^...
CooLee's user avatar
  • 375
1 vote
0 answers
270 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
Madhuresh's user avatar
  • 157