Tagged Questions

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

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3
votes
1answer
120 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. ...
4
votes
0answers
95 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 = ...
2
votes
0answers
68 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 ...
1
vote
1answer
64 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 ...
4
votes
1answer
229 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 ...
1
vote
0answers
96 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 ...
1
vote
0answers
43 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 $ ...
1
vote
0answers
92 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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0answers
43 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 = ...
2
votes
1answer
214 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 ...
4
votes
1answer
115 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 $V:\mathbb{R}^3\to \mathbb{R}$ ...
0
votes
0answers
200 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 ...
1
vote
0answers
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 ...
2
votes
0answers
121 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 ...
2
votes
1answer
149 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 $A$ ...
1
vote
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 ...
2
votes
2answers
83 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 ...
2
votes
1answer
40 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 ...
0
votes
1answer
140 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 ...
5
votes
1answer
204 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 ...
0
votes
0answers
376 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 ...
2
votes
1answer
104 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 ...
3
votes
1answer
221 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 $T$ ...
2
votes
1answer
282 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 ...
1
vote
1answer
70 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 ...
1
vote
1answer
238 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 ...
0
votes
1answer
307 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 ...
2
votes
0answers
211 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 ...
2
votes
0answers
37 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 ...
2
votes
1answer
116 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 ...
3
votes
0answers
270 views

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 ...
2
votes
2answers
863 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, $x$ ...
4
votes
0answers
49 views

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 ...
0
votes
2answers
91 views

Trace of a diagonalized matrix

Why do I have: $Tr(SDS^{-1})=Tr(D)$?
0
votes
4answers
65 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?
1
vote
1answer
45 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 ...
0
votes
1answer
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 ...
1
vote
0answers
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 ...
5
votes
1answer
153 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 $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
0
votes
0answers
57 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 ...
1
vote
0answers
50 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 ...
0
votes
3answers
103 views

The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$

On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
2
votes
1answer
93 views

Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
1
vote
2answers
79 views

Upper bound on the difference between two elements of an eigenvector

Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
0
votes
1answer
71 views

The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Suppose $x$ is invertible in the unital Banach algebra $A$. How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
0
votes
1answer
48 views

Spectrum of T in $B(\ell^2)$

Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows: $$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$ What is spectrum of T in $B(\ell^2)$?
2
votes
0answers
67 views

Calculus of Variations statement of a Singular Value Decomposition?

My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
3
votes
0answers
85 views

SVD, infinite matrices and normal operators from a function

I'm trying to understand the behavior the Singular Value Decomposition on a deeper level, and why it might give a particular result. Take the function $$ f(x,y) = \frac{1}{(1+2x+y)^2} $$ and ...
1
vote
0answers
34 views

What is twisted triangular two-torus also called a triangular doughnut?

In "A Geometry of Music" by Dmitri Tymoczko Oxford 2010, the authors says that mathematicians refer to a particular lattice in what mathematicians would call "the interior of a twisted triangular ...
5
votes
1answer
101 views

Gelfand's formula, different field

Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal ...