Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
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A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
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A Spectrum of a compact operator in $\ell^p$

Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$ T((x_1, x_2, \ldots)) = (\alpha_1 ...
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Characterization of compact operators by their spectra

In any functional analysis book there is usually a section devoted to the study of the properties of the spectrum of compact operators. Is there any spectral characterization of compact (self-...
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1answer
36 views

Non-Isometric of the Wolpert's problem

Related to the document Drums That Sound the Same of S. J. Chapman, he has proved those two shapes was isopectrals. Question : I don't know how to prove that they are not isometric (Riemannian ...
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Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
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80 views

First derivative of the transposition to be continuous

In the document That Sound the Same by S. J. Chapman page $127$, he explains the following concept : In order for the first derivative of the tranposition to be continuous it is sufficient that ...
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Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
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Why is the Minimum in the Min-Max Principle for Self-Adjoint Operators attained?

Let's consider a self-adjoint operator $A$ (not necessarily bounded) on a Hilbert space which is bounded from below, with domain $D$ and whose resolvent is compact. Then, the spectrum consists solely ...
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Numerical analysis of bifurcation problems - notational confusion

I'm going through the following set of review notes about numerical analysis of bifurcation problems and was wondering if someone could explain to me what is going on on the page 50. First of all, I'...
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63 views
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How linear map transform the unit ball?

Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ ...
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1answer
52 views

Why $V_i = \mathcal{P}_i (M,g)$?

In the book Le Spectre d'une Variété Riemannienne of Berger, Gauduchon and Mazet, they explain the following : Let $(M,g)$ a Riemannian manifold, and suppose for all $i \in \mathbb{N}$ a vector ...
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When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where \begin{equation} P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
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1answer
43 views

Why is the Maximum in the Min-Max Principle for Self-Adjoint Operators attained?

Let's consider a self-adjoint operator $A$ (not necessarily bounded) on a Hilbert space which is bounded from below, with domain $D$ and whose resolvent is compact. Then, the spectrum consists solely ...
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41 views

Symmetries and eigenvalues of the Laplacian.

Lets consider a domain $\Omega \subseteq \mathbb R^2$ smooth enough, and the eigenvalue for the laplacian \begin{align} -\Delta u &= \lambda u &x\in\Omega\\ u &= 0 &x\in \partial \...
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Surjectivity of $\Delta : \mathcal{P}^k \to \mathcal{P}^{k-2}$?

Let $P^k$ be the space of homogeneous polynomials of degree $k$, i.e. $P^k = \text{span} \{x_1^{k_1} \dots x_n^{k_n} : k_1 + k_2 + \dots + k_n = k\}$. I am trying to show that the Laplacian operator $\...
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Prove that $p$ is a differentiable covering and a local isometry

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori. The lattice acts on $\mathbb R^n$ by $$γ(x)...
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Questionning of the resolution of the Tori

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori. The lattice acts on $\mathbb{R}^n$ by $$\...
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32 views

Convergence of a Sequence of Continuous Functions of Bounded Operators

Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, $\{A_n\}_n$ a sequence of self-adjoint operators in $\mathcal{B}\left(\mathcal{H}\right)$ (the bounded linear operators on $\mathcal{H}...
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1answer
128 views

How could I find an orthogonal basis for $H^k$? - HELP!!

Let $P^k=$homogeneous polynomials of degree $k$ in $x$, $y$, $z$, $k=0, 1, 2, \dots, $ i.e. $P^k= \text{span} \{x^{k_x}y^{k_y}z^{k_z} : k_x+k_y+k_z=k\}$ and $H^k= \{f \in P^k : \Delta f = 0\}$, where $...
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Compact resolvant inequality

I want to prove that if an operator $A$ with domain $D(A)=\left\{u\in L^2\;\text{such that}\; Au\in L^2(\mathbb{R}^n) \right\}$ has a compact resolvant then there exist a constant $c>0$ such that ...
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52 views

Finding this operator's spectrum

In an exam, my professor gave the following exercise: State and prove the spectral theorem for compact operators. Let $K$ be the operator defined by: $$Kf(t)=\int_0^1\min(t,s)f(s)\mathrm{d}s.$...
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$H^k= \{f \in P^k : \Delta f = 0\}$?

Let $P^k=$homogeneous polynomials of degree $k$ in $x$, $y$, $z$, $k=0, 1, 2, \dots, $, i.e. $P^k= \text{Span} \{x^{k_x}y^{k_y}z^{k_z} : k_x+k_y+k_z=k\}$. This is maybe a silly question, but I am not ...
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Symmetry of the problem in using the group of rotation of $\mathbb{R}^3$

In Spectral Geometry in Non-standard Domains page $37$ point ($2$), it discusses of the symmetry of the problem in using the group of rotation of $\mathbb{R}^3$ with certain properties. Could anyone ...
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1answer
25 views

Properties of matrix stable (numerical) rank

I happened to notice that there is concept "stable rank" that people used a lot in matrix computation theories, such as the work of Rudelson & Vershynin (2005). It is defined to be the ratio ...
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If $\,\Delta_{\scriptsize\mathbb{R}^n}u = \lambda u\,$ and $\,u(x)=u(x+y)$, then $\,u_{y}(x):=e^{2 \pi i \left\langle x,y\right\rangle}$

Related to Analysis on Manifolds via the Laplacian page $52$, I would like someone explain to me why if we have a function $u$ such that $\,\Delta_{\mathbb{R}^{n\,}}u = \lambda u\,$ and $u(x)=u(x+y)$ ...
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$\lim_{k \to \infty} \frac{N(\lambda_k)(2 \pi)^3}{\omega_3 \text{Vol}(\mathbb{S}^2) \lambda^{\frac{3}{2}} } =1 ~\not = 0$

I would like to use Weyl's to confirm my result from the spectrum of $\mathbb{S^2}$. So far I found that the spectrum of $\mathbb{S^2}$ is $\{k(k+1) : k \in \mathbb{N} \cup \{0\} \}$ and each ...
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49 views

A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
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81 views

Min-Max principle - Subspaces over $H_0^1$

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem $$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\...
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2answers
276 views

Min-Max Principle with matrices - Understanding $\lambda_2$

Related to the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I am trying to do the same with $$A= \begin{bmatrix} 2 & 0 & 0\\ 0 &...
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Extending a unitary operator

Suppose that $\mathcal{H}_1$ and $\mathcal{H}_2$ are two separable Hilbert spaces and that $X\subset \mathcal{H}_1$ is a dense subspace (i.e. $\overline{X}=\mathcal{H}_1$). If $\operatorname{W}:X \to \...
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$\displaystyle\frac{\langle A x_{\min}, x_{\min}\rangle }{\langle x_{\min}, x_{\min}\rangle }=\lambda_1 \not = \lambda_2$? - Explanations

Related the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I tried to complete the details of mickep's answer, but I have a little problem....
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Complex dispersion relation - example

I am dealing with eigenvalue problems of linear operators $L$ on the real line that can be written as a linear nonautonomous ODE $$ v_{\xi}=A(\xi;\lambda)v,~~\xi\in\mathbb{R},\lambda\in\mathbb{C}, v\...
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operator inequality using spectral theorem

Given two densely defined unbounded self-adjoint strictly positive operator $A$ and $L$ in Hilbert space $H$ with domain $D(A) \subset D(L)$ and $\|Lx\| \leq \|Ax\|$ for all $x\in D(A)$, why do we ...
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Compactness of the resolvant of a Schrodinger operator

I want to prove that the closure of an operator $A$ with domain $D(A)= \mathcal{C}^{\infty}_c(\mathbb{R}^n)$ given by: $A=\Delta+\frac{1}{4}||\nabla V(x)||^2-\frac{1}{2}\Delta V(x)$ has a compact ...
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1answer
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A restriction $A_2$ of a compact self-adjoint compact linear operator $A$ is also compact and self-adjoint?

Let $X$ be an inner product space and let $A$ be a compact and self-adjoint linear operator. Let $p_1$ be an eigenvector of $A$. Let $A_2$ be the restriction of $A$ to $X_2$ where $X_2$ is given by $$...
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Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
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1answer
24 views

Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
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Ineqality involving operators

I need your help to solve the following problem: We define the operatorss $a$ and $b$ with domain $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$ by: $a=\partial_x+\frac{1}{2}\partial_xV(x)$ (where $V$ in ...
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Neumann vs Dirichlet eigenvalue problem - Intuition

What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, ...
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Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations

In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ...
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2answers
25 views

Question about Langevin equation

The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
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1answer
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Domain Monotonicity - Neumann eigenvalue problem (Edit)

Related to the question : http://mathoverflow.net/questions/242136/why-m-1-subset-m-2-not-rightarrow-n-m-1-lambda-leq-n-m-2-lambda The Neumann eigenvalues of the rectangle with sides $a$ and $b$ are $...
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1answer
48 views

Spectrum of $T\in \mathcal{L}(E)$, such that $T^n=I$

Let $T:E \to E$ be a bounded linear operator, $E$ infinite dimensional Banach space, such that $T^n =I$, for $n\ge2.$ Show that $\sigma(T)\subset\{-1,1\}.$ My idea is show that $\|T\|=1$ initially,...
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Is this a general identity for the Resolvent? [solved: integral representation of the resolvent]

And if so, what is it called? $$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$ as in Reed-Simon XIII.7 example 1. It is stated there for $H=-i\...
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Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
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1answer
24 views

Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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1answer
36 views

Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
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25 views

Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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1answer
64 views

How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law

I am trying to prove the Weyl's asymptotic law for eigenvalues. In the document Weyl's law of p. $4$, I have managed to go up to the step $$\tilde{\nu_k} \leq \nu_k \leq \mu_k \leq \tilde{\mu}_k \...