Tagged Questions

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

learn more… | top users | synonyms (1)

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
155 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
60 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
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 ...
0
votes
3answers
104 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
80 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
73 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
50 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
68 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
106 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 ...
2
votes
0answers
90 views

Dirichlet eigenvalue problem on the Hilbert cube

I'm trying to solve the Dirichlet problem for the Helmholtz equation \begin{aligned}-\triangle u & = & \lambda u, & x\in\Omega,\\ u & = & 0, & x\in\partial\Omega, ...
3
votes
1answer
507 views

Spectral Theorem for bounded compact, self-adjoint operators as corollary of Hilbert-Schmidt theorem

I'm following Debnath and Mikusinksi's "Introduction to Hilbert Spaces with Applications" and am trying to understand how the spectral theorem for compact self-adjoint operators is a corollary of the ...
2
votes
1answer
208 views

spectrum of unitary operator

On $L^2(-\infty, \infty)$, T is a bounded linear operator and is defined by $$Tf(t)= \begin{cases} f(t), & \text{for } t \geq 0 \ \cr -f(t), & \text{for } t<0, \end{cases}$$ I'm able to ...
4
votes
1answer
138 views

Spectrum proofs

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
1
vote
1answer
149 views

Spectral family property: $\lambda \geq M \Rightarrow E_{\lambda} = I$

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and I'm trying to follow his proof about some properties of a spectral family associated with a bounded self-adjoint ...
2
votes
1answer
81 views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
0
votes
2answers
160 views

Representation of a bilinear form on an Hilbert space

Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated. 1) There exists a symmetric ...
2
votes
1answer
93 views

Null spaces and projections

I'm following Kreyszig's exposition of projections in "Introduction to Functional Analysis with applications". I'm trying to follow the proof of the following theorem (9.6-1 on pp. 486-487) regarding ...
2
votes
1answer
252 views

Spectrum of the unbounded operator $i\partial_x$

I've been puzzling over this for some time now, and can't quite make my intuitions precise. I need to find the resolvent set and spectrum of the operator $$ Lu=i\frac{du}{dx} $$ taken to be ...
1
vote
1answer
174 views

Residual spectrum is empty

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468) For a bounded self-adjoint linear operator ...
2
votes
1answer
78 views

Can I deal with the weak derivative in the “strong” sense?

This is an exercise in functional analysis: For $k=1,2,3$, let $A_k: D(A_k)\subset L^2([0,1])\to L^2[(0,1)]$ be the first-order differential operators $A_ku=iu'$ with domains $$ D(A_1) = ...
1
vote
0answers
84 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
1
vote
0answers
138 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
3
votes
1answer
168 views

Convergence of operator norm

I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
5
votes
2answers
114 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
4
votes
1answer
105 views

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
4
votes
1answer
254 views

Study of the Laplacian on the Hyperbolic plane

What's a good reference for the simplest case? I'm interested in the spectral theory of the Laplace-Beltrami operator on the upper half plane (domain, self-adjoint extension, etc.). I only need this ...
1
vote
1answer
59 views

Showing a bound on a contour integral

I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
1
vote
1answer
65 views

Proof that $A^n = \frac{1}{2\pi i} \oint _C z^n \left(z - A \right)^{-1} dz$

I'm working my way through Martin Schechter's 'Principles of Functional Analysis' (2nd ed.) and am trying to understand his proof of the following theorem, given on page 136: "Let $T:X\to X$ be any ...
2
votes
1answer
105 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
0
votes
1answer
97 views

real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
2
votes
1answer
66 views

Eigenvectors of a Self-adjoint Differential Operator Spans the Domain

Can someone, please, suggest a reference or what steps should I take to prove the following theorem: The set of eigenvectors of a self-adjoint differential operator, defined over a finite ...
2
votes
1answer
120 views

Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
1
vote
1answer
185 views

Continuous functional calculus

Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ ...
1
vote
1answer
78 views

Convergence of Fourier series - strange graph in proof

I am reading a text that states the following related to convergence of Fourier series: $$g_K(x) = > ...
2
votes
1answer
113 views

Global solution for spectral clustering

I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
1
vote
1answer
128 views

Spectral density function in stationary process

I have the following argument in my Time Series class notes: Let ${u_t}$ be a mean zero covariance stationary process. Define $\gamma(j) = \mathbb{E}u_tu_{t-j}$ and $Y_t = \mu +C(L)u_t$ where ...
3
votes
1answer
253 views

Spectral theorem of compact operators in Hilbert space

I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof. Theorem. Let $H$ ...
0
votes
1answer
131 views

Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 ...
3
votes
1answer
79 views

Spectrum of the Hill Operator $L(y)= -y''+ v(x) y $

Consider the eigenvalue equation for the Hill operator $$L(y)= -y''+ v(x) y = \lambda y, \quad x\in \mathbb{R},$$ where $v(x)$ is any potential and $\lambda$ is the spectral parameter. If $v(x) ...
0
votes
2answers
263 views
0
votes
0answers
76 views

Spectral theory - a simple application

I'm following Martin Schechter's 'Principles of Functional Analysis' (Second Edition, 2002) and am interested in the spectral theory chapter (chapter six). In particular, I wish to make us of ...
3
votes
1answer
56 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
3
votes
0answers
68 views

In relation with the set of Fredholm perturbation elements

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 ...