Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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Using Weyl sequences to prove relation between quadratic form and spectral radius

I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$ holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula ...
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25 views

Time Series: Spectral Density

Prove that $$ \int_{-\pi}^\pi e^{i(k-h)λ} dλ = 0 $$ if k!=h solving the integral gives $$ \frac{2 sinh(\pi * i(k-h))}{i(k-h)} $$ I don't see how this will give 0 with k!=h (also if k=h, the ...
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122 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
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132 views

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ...
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128 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
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37 views

Do eigenfunctions from symmetric ODEs really make up a base for $L^2$?

I'm confused about a theorem in some hand-out material on spectral theory for ODEs. The BVP is stated in the form $$ \begin{cases} L u = f\\ R u = 0 \end{cases} $$ over $I = [a,b]$. And ...
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66 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
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62 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
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106 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
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83 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
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64 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
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1answer
62 views

Asymptotic of the heat kernel

I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian on a Riemannian Manifold" and not quite clear how to get the estimate $(4\pi t)^{n/2}|Q_k * H_k|\leq C \cdot t^{k+1}$ on a compact ...
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225 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
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62 views

Spectral radius of nonnegative matrices with an M-matrix difference

Let $A$ and $B$ be non-negative matrices (both with real spectra) and $A−B$ is an M-matrix. Are these conditions necessary to prove that the spectral radius of $A$ is greater than the spectral radius ...
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95 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
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42 views

Solving an inverse spectral problem

In order to solve the inverse spectral problem: $$ -y''(x)+q(x)y(x)= \lambda _{n}y(x) $$ If we want to obtain $ q(x) $ what we should need about the spectrum? a) The eigenvalue staircase $ ...
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89 views

Examples of deeper results in finite-dimensional vector spaces?

this one is a bit inverted! So I am busy doing an advanced undergrad course in Linear algebra, and it is going very well, the problems in the book seem fairly routine. To be able to see if I am any ...
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40 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
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192 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
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1answer
113 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
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158 views

How does determinant relate to argument of matrix function?

I have a matrix function $R \mapsto J(R)$ from $\mathbb{R}$ to the set of irreducible matrices with non-negative entries. We can assume that $J(R)$ is $d \times d$, although any solutions that work ...
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130 views

Symmetry adapted basis function to make the Hamiltonian matrix Block Diagonal.

Can anybody give me a tip to solve this problem? I have large quantum mechanical Hamiltonian, to solve it numerically I have to decompose it into the block diagonal form. To convert the hamiltonian ...
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15 views

Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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105 views

Application of the spectral decomposition theorem to PDE

In the compact version, there is many application of the spectral decomposition of a bounded self-adjoint operator: Sturm-Liouville, spectre of the laplacian,... But for the general version which ...
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138 views

Show that an operator is bounded (from Reed and Simon)

I am currently reading Reed and Simon's IV: Analysis of Operators, Volume 4 (Methods of Modern Mathematical Physics). I don't understand something they do in Theorem XIII.64. The problem is: Let $A$ ...
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1answer
46 views

The trace of an integral equation?

I am reading a paper about spectroanalysis and encountered the following integral equation: $$\int_{-1}^{1}\frac{\sin A(x-x')}{\pi(x-x')}\psi(x')dx'=\lambda\psi(x)$$ Then the paper gives without proof ...
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2answers
75 views

Spectrum of an element

I'm having a little trouble calculating the spectrum of an element: specifically, the element $f(x) = \frac{1}{x}$, as an element of the bounded continuous functions from $[1, \infty)$ with pointwise ...
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1answer
38 views

Spectrum and tower decomposition

I'm trying to read "Partitions of Lebesgue space in trajectories defined by ergodic automorphisms" by Belinskaya (1968). In the beginning of the proof of theorem 2.7, the author considers an ergodic ...
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129 views

Prove a bilinear operator is symmetric and positive definite

I'm having problem showing the following: All operators are defined on $V$ which is real (not complex). Let $f$ be a bilinear operator that is anti-symetric (meaning $f(a,b)=-f(b,a))$ and let $J$ be ...
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193 views

How to prove that the spectrum of the Laplacian over $\Omega\subset \mathbb{R}^n$ is negative?

I am looking for a proof of this well known fact and I guess it has to do with integration by parts (Green's identity). Unfortunately, I only know about 1-d integration by parts( I am just 3rd ...
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353 views

Spectral Theorem for Commuting Self-Adjoint Operators

Welcome everybody :) I'm working together with a little group preparing for the upcoming exams in "Mathematical Methods of Physics." There's one tricky task involving some linear algebra, namely the ...
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1answer
100 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
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203 views

Simple spectrum and the spectral theorem for bounded symmetric operators

I have a question regarding the spectral theorem for bounded self-adjoint operators. The book "Functional Analysis, an Introduction" by Eidelman, Milman, and Tsolomitis says that if an operator $T$ ...
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236 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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67 views

Resolvent lemma

I would like to proof a lemma that I am quite sure should be correct as I found it somewhere, I am writing a thesis about quantum walks and need this to get through an article. Let $X$ be Banach ...
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1answer
212 views

Bounded operator inverse, norm and spectrum

I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator: $X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$ 1. I need to show that an ...
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226 views

$uu^T$ is the standard matrix for the orthogonal projection of $\mathbb{R}^n$ on the subspace spanned by $u$

Let $A$ be an $n\times n$ symmetric matrix, and $u$ an eigenvector of $A$. Why is it true that $\forall x\in\mathbb{R}^n$, $x^{T}uu^{T}(I-uu^T)x=0$? If I'm able to show this is true then I can show ...
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204 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
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35 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
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107 views

Representation of square-integrable kernel

Let $k: [0,T] \times [0,T] \to \mathbb{R}$ symmetric, square-integrable and define $$(Kf)(t) := \int_0^T k(s,t) \cdot f(s) \, ds \qquad (f \in L^2([0,T])$$ Since $K$ is compact, by the spectral ...
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spectral integral

I am learning spectral integration for my summer. I am stuck at a point. Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any ...
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746 views

Eigenvectors of inverse complex matrix

For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
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Is it possible for an operator to have only one eigenvalue in this case? - in need of a proof

First of all i have to state that i am a newcommer to spectral theory so please take it easy on me :). On lectures our professor derived this equation: \begin{align} \underbrace{\psi ...
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76 views

Trace of a diagonalized matrix

Why do I have: $Tr(SDS^{-1})=Tr(D)$?
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4answers
61 views

Eigenvalues of power of matrices

How come if $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$? And is its multiplicity necessarily the same?
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1answer
42 views

Largest eigenvalue of a graph

I have $\lambda_1$ the largest eigenvalue of a graph, with $x = (x_v)_{v \in V(G)}$ the corresponding eigenvector. $x_u$ is the entry of $x$ with maximum absolute value. I don't understand why I ...
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23 views

Eigenvectors orthogonal to $j$

I'm studying the proof of the following statement: $Spec(K_n) = (n-1)^1(-1)^{n-1}$ At some point I have: By the Spectral Theorem, when looking for eigenvectors $v$ we can assume they are ...
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26 views

looking for “invertibility and singularity”

Dear *friends* Many monts ago,i searched a lot the book of Robin Harte "invertibility and singularity". this book contains a lot of demonstrations that i need in my master. it focus on banach ...
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1answer
144 views

Self adjointness of an elliptic differential operator

Let $A:D(a)\to L^2(\mathbb{R}^n)$ be an elliptic partial differential operator $$ A(f)=\sum_{i,j=1}^{\infty}\partial_{x_j}(a_{ij}(x)\partial_{x_i}f) $$ where $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
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52 views

Resolution of Identity

I am reading the twelfth chapter of Rudin's Functional Analysis and am unable to understand two comments he makes in passing in the section on Resolution of Identity. Let E be a resolution of identity ...