Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.
1
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0answers
13 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
29 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
11 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 ...
4
votes
0answers
30 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
39 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,
...
2
votes
1answer
46 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 ...
0
votes
0answers
15 views
Joint Probability in terms of Spectral Matrix
Given a multivariate time series ${\bf x}(t)$ with known spectral matrix ${\bf S}(f)$, how would I be able to display the likelihood function
$$p_{{\bf x}}({\bf x}(1),{\bf x}(2) \dots {\bf x}(N))$$
in ...
2
votes
1answer
41 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 ...
2
votes
1answer
43 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
18 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
48 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 ...
1
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0answers
34 views
Continuous spectrum for a specific linear operator
The operator is for $A:L^2[-1,1]\to L^2[-1,1]$ defined via
$$Au(x)=xu(x)+\theta\int_{-1} ^1u(s)ds.$$
The question is actually find the spectrum, but I managed to find everything else, pending the ...
0
votes
1answer
68 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
45 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
37 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
34 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
31 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
$$
D(A_1) = ...
1
vote
0answers
25 views
Find spectrum of the operator and an explicit form for the solution
Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if
$v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$
(a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum
(b) ...
1
vote
0answers
32 views
Find spectrum of the operator
I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem.
Let $0 \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator
$Au=v,$ where ...
3
votes
1answer
67 views
Convergence of operator norm
I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
5
votes
2answers
47 views
Looking for a first order perturbation of the Laplacian having 0 in its spectrum
I would like to find an explicit example of a linear elliptic operator having the following form:
$$Lu=-\Delta u +b(x)\cdot \nabla u, $$
where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
3
votes
1answer
39 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
31 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 ...
0
votes
0answers
12 views
Spectra of composition of graphs (lexicographic product)
I would like to know how to relate the eigenvalues (eigenvectors) of the lexicographic product of two graphs in terms of the eigenvalues (eigenvectors) of the factors...I hope someone can help me...
1
vote
1answer
28 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
41 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
84 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.
-1
votes
0answers
14 views
left,right,and boundary
let A a banach algebra .
let x is an element of A
let denote σ(x) the spectrum of x ,
σleft(x) the left spectrum of x
σright(x)the right spectrum of x
∂σ(x) the boundary of σ(x)
i need help to ...
0
votes
1answer
33 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
34 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
61 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
47 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$ ...
0
votes
0answers
32 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
73 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
28 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
109 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
0answers
18 views
spectral representation of discrete time, periodic, weakly stationary sequence
Let $(\xi_n)_{n\geq 1}$ be a sequence such that $\xi_{n+N} = \xi_n$ for some $N$ and all $n$.
What would be the spectral representation of this sequence?
Let $b(t)$ be the covariance function for ...
0
votes
1answer
57 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
43 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
63 views
Are any matrices positive semidefinite, non-negative, and not diagonally dominated?
If so, I'd appreciate any examples. Thanks.
0
votes
0answers
35 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
41 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
$$\{(r, θ) : 0 ≤ θ < 2π, ...
3
votes
0answers
59 views
In relation with the set of polynomially Fredholm perturbation elements
Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of polynomially Fredholm perturbation elements in $A$, i.e.
...
1
vote
0answers
46 views
Determining the spectral density?
Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and ...
0
votes
1answer
30 views
The spectrum of $C(K)$ where $K$ is a compact Hausdorff space
Let $K$ be a compact Hausdorff space, what's the spectrum of $f\in C(K)$? I don't know how to start.
2
votes
0answers
36 views
What are the connections between spectral expansion and differential operator?
For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion,
$$f(x)=\sum_n \hat{f}(n) e^{inx},$$
where the exponentials are eigenfunctions for differential operator ...
2
votes
2answers
79 views
Eigenvalues of $A^{T}A$
Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose.
Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
2
votes
1answer
50 views
Spectrum property
Can someone please provide good demonstration for this theorem:
Theorem 1.9. Let $A$ be a Banach algebra. If the elements $a, b \in A$ satisfy
$ab = ba$, then
$\sigma (a + b) ⊆ \sigma (a) + \sigma ...
0
votes
0answers
30 views
Unital algebra and its spectrum
It is two of a list of questions in one problem.
If A is a unital algebra and $S\subset A$, define $S'=\{a\in A:ax=xa,\forall x\in S\}$, how to prove that
1)$S''$ is commutative if $S$ is ...
1
vote
2answers
125 views
Spectral radius of the Volterra operator
The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$
How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
