Questions tagged [spectral-theory]

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

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Eigenvalue problem on the real line

The following is a problem in a text (in Portuguese) on Critical Point Theory that I am reading: Find the eigenvalues and eigenfunctions of the problem $$ (P) \quad \begin{cases} - y'' = \lambda ...
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essential spectrum of self-adjoint extensions

I read in Weidman's book that "If $T_0$ is an operator with deficiency indices (m,m), then all self-adjoint extensions have the same essential spectrum " (page 163). I guess the reason is that the ...
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The rank of a spectral projection $E_A(\Delta)$ when $\Delta \bigcap \sigma_e(A) \neq \emptyset$

Let $H$ be a infinite dimensional Hilbert Space. $A \in B(H)$ and $A$ self-adjoint. Let $\sigma_e(A)$ be the essential spectrum of $A$. Since $A$ is self-adjoint, we can assume the convex hull of $\...
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Does $\|A\|$ or $-\|A\|$ belongs to $\sigma(A)$ for a normal operator

For a normal operator $A$ on a Hilbert space $H$, I know that spectral radius $r(A)=\|A\|$. The question says is it true that either $\|A\|$ or $-\|A\|$ belongs to spectrum $\sigma(A)$? I have ...
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On the resolvent set of an unbounded operator

Suppose $A$ is the infinitesimal generator of a $C_0$ semigroup $S(t)$ on an Hilbert space $X$. If $$\langle Ax, x\rangle \leq \omega \|x \|^2 \ \ \ \forall x \in \mathfrak{D}(A)$$ then $$\|S(t)\...
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Computing eigenspectrum of $T:L^2[0,1]\to L^2[0,1]$, defined by $(Tf)(t)=tf(t)$

I have to compute the eigenspectrum of operator $T:L^2[0,1]\to L^2[0,1]$, defined by $(Tf)(t)=tf(t)$. If $\lambda$ is an eigenvalue, then there is a nonzero $f$ such that $Tf=\lambda f$ $\Rightarrow$ $...
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Proof of $\lambda \in \sigma (T)$ implies $|\lambda|\leq ||T||_{B(X)}$

I want to show that $\lambda \in \sigma (T)$ implies $|\lambda|\leq ||T||_{B(X)}$ where $\sigma (T)$ is the spectrum of $T$ and $T\in B(X)$. I would like to check my proof here as it is different and ...
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Two commuting self-adjoint unbounded operator over a Hilbert space

Let $A,B$ be two commuting unbounded self-adjoint operators over a Hilbert space $H$, then there exists a measure space $(X,\mu)$, and isometry $H\cong L^2(X,\mu)$, such that $A,B$ act on $L^2(X,\mu)$ ...
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Eigenspace of linear completely continuous field

Let $H_1$ and $H$ be Hilbert spaces, and $H_1 \subset H$ be a dense and compact inclusion. A linear operator $L: H_1 \to H$ is called a completely continuous field if $L = - A + B$, where $A: H_1 \to ...
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I need to find the spectrum of an operator.

I need to prove that the spectrum of operator A in $L_{2}[0,1]$ is [-1;1] where $Ax(t) = \sin(\frac{1}{t})x(t)$ if $ t > 0$ and$ Ax(0) = 0$ Elementary - norm of $A$ is less or equal to $1$. Hence, ...
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Spectral properties of hypercyclic operators

I'm studying some topics related to the invariant subspace problem, and consequently I find myself dealing with hypercyclic operators. (An operator $T:X\rightarrow X$ is hypercyclic if there is some ...
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Prove that a normal matrix can be completely determined by its eigen values and a unitary matrix.

Prove that a normal matrix can be completely determined by its eigenvalues and a unitary matrix. I tried using the Spectral decomposition theorem: Is states that a normal matrix N with spectrum set {$...
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64 views

Self adjoint extensions

I'm taking a course in Functional Analysis using some topics from Kreyszig and Reed & Simon books, I have been asked to solve the following exercise: Let $A$ be a symmetric operator such that $\...
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Spectrum of $Tf(x)=\int\limits_{-\infty}^\infty \frac{f(y)\,dy}{1+(x-y)^2}$

How can one find spectrum of this $T: L_2(\mathbb{R})\to L_2(\mathbb{R})$? I kinda hoped that this operator is compact, so that I could look only for point spectrum, but $K(x,y)\notin L_2(\mathbb{R}^...
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I need to prove that the spectrum of operator A in $L_{2}[0,1]$ is [-1;1] where $Ax(t) = \sin(\frac{1}{t})x(t)$ if $ t > 0$ and$ Ax(0) = 0$ [closed]

Elementary - norm of $A$ is less or equal to $1$. Hence, all $\lambda$ from spectrum is such that $|\lambda| \leq$ 1. $g(t)$ := $\sin(\frac{1}{t})$ if $t > 0$ and $0$ if $4t = 0$. If there exists $...
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Some questions about wave operator.

On sec. 3.4.1 in Schlag & Nakanish: invariant manifolds and dispersive Hamiltionian evolution equations, the authors talked something about wave operators. The wave operators are defined as the ...
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Laplacian and eigenvectors relationship

I have a Laplacian matrix $L_G$ of a connected and undirected graph $G$. $L_G$ is symmetric, positive semi-definite. I have another laplacian $L_H$, that is also symmetric and positive semi-definite, ...
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References about properties of the spectra of the Laplace-Beltrami on p-forms over homogeneous spaces

I am reading the paper "Specra and Eigenforms of the Laplacian on $\mathbb{S}^n$ by Ikeda-Taniguchi and $\mathbb{P}^n(\mathbb{C})$" and i want to know where do i can get a proof or some ideas to prove ...
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28 views

Let $U \in U(H)$ be unitary operator on Hilbert space $H$. Is it possible for it to has an empty point spectrum?

Let $U\in U(H)$ be unitary operator. Is it possible for it to has an empty point spectrum? I am aware that every bounded operator acting on complex Hilbert space has non-empty spectrum. Since $\...
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Show that there exist $a_1,\ldots, a_{2n-1}$ such that $ a_{2n-1}J^{2n-1}+\cdots+a_1 J=I_n,$ where $J$ is a Jordan matrix

Let $J\in\mathbb{C}^{n\times n}$ be a Jordan normal form and assume that ${\rm tr~}J<2n$. Prove or disprove that there exist $a_1,\ldots, a_{2n-1}\in\mathbb{R}$ such that \begin{equation} a_{2n-1}...
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Is this a characterization of the resolvent?

I am trying to understand a statement that is in some notes that I am reading right now. It is the following. "Let $T$ be a bounded, self-adjoint operator, $\eta\in\mathbb{R}, \eta\neq 0$ and let $H$ ...
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Does spectral approximation imply approximation in trace?

Suppose $B\in\mathbb{R}^{m\times n}$ is an $\varepsilon$-spectral approximation to $A\in\mathbb{R}^{m\times n}$, that is, for all $x\in\mathbb{R}^n$, $$(1-\varepsilon)||Ax||_2^2\le ||Bx||_2^2\le (1+\...
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Beauty of Spectral Graph Theory

Why would one choose to study spectral graph theory? Where can the spectrum of complete graphs, for example, be applied in real- life example or of any graph in general? A brief historical background ...
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37 views

$X$ Banach, $T(X\to X)$ with compact resolvent implies $T+B$ is Fredholm with index $0$ provided $B$ is bounded

I am trying to solve this problem from Gohberg's book: Let $X$ be a Banach space and suppose that $T : \mathcal{D}(T) \subset X \to X$ has compact resolvent. Then, if $B$ is bounded on $X$, the ...
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29 views

Fourier coefficients of Eisenstein series

Several questions/reading reference requests for the following topics. I require some point-wise bounds on the absolute value of the Eisenstein series $E_{\mathfrak{a}}(\sigma_{\mathfrak{b}}z,1/2 + ...
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41 views

Spectrum of a $\ell^2$ operator and compactness

Let $T: \ell^2 \to \ell^2$ defined as follows: $T(x_1, x_2, x_3, x_4, x_5.., x_n,..) = (0, x_1, 0, x_4, x_5 .., x_n,..)$. Find the spectrum of $T$, the eigenvalues of $T$ (if they exist) and the ...
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Unitarily equivalent multiplication operators

Let $A$ be the operator given by $Ax(t)=\sin(t)\,x(t)$ in $L_2[0,2\pi]$, and $B$ the operator given by $Bx(t)=\sin(t)\,x(t)$ in $L_2[-2\pi,2\pi]$. Are these operators unitarily equivalent?
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Decomposition of compact operators

Giving an infinite dimensional Hilbert space $H$ and a compact operator $K$ on $H$ with spectrum $\sigma \left( K\right) =\left\{ 0,\lambda _{0},\lambda _{1},\ldots \right\} $, I'm trying to prove ...
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32 views

Compare of eigenvalues $λ_{1,\Omega_2}$, $λ_{1,\Omega}$ and $λ_{1,\Omega_1}$

Recall that $\lambda_1$ is the smallest eigenvalue of the Laplacian with boundary conditions of Dirichlet and we know that the link between $\lambda_1$ and the smallest possible constant in the ...
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Approximate point spectrum $ T(x_1, x_2, \ldots) = (x_2, x_3, \ldots), \quad x \in \ell^p.$

Let $T:\ell^p \to \ell^p, \quad \text{for }1\leq p\leq \infty$ defined as $$ T(x_1, x_2, \ldots) = (x_2, x_3, \ldots), \quad x \in \ell^p.$$ We defined the approximate point spectrum $T$ as the ...
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What is the spectrum of the continuous extension of a bounded operator?

If $H$ is a Hilbert space and $H_{0}$ is a dense subspace in $H.$ Giving a bounded operator $A_{0}:H_{0}\rightarrow H_{0}$ with $\sigma \left( A_{0}\right) $ as spectrum, what would be $\sigma \left( ...
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How to calculate a multiplication operator representation?

Let $H =\ \mathcal{l}^{2}(\mathbb{Z},\mathbb{C})$ and $A = R + L$, where $L$ is the left-shift operator (and $R$ is the right-shift $(Ra)_{n}=a_{n-1}$). Set $$U : \mathcal{l}^{2}(\mathbb{Z},\...
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63 views

The Spectrum of the operator in C[0;1]

I have an operator $ Ax(t) = \int_{0}^{t^2} x(s)ds $ in $ C [0;1] $. I need to find spectrum of this operator. $ A $ is a compact operator, so the spectrum consists of 0 and eigenvalues. As I ...
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Fractional Laplacian maps positive functions into positive functions?

Assume $f \geq 0$ is $C^\infty$ with compact support. Is it true that $$(-\Delta)^\alpha f \geq 0$$ where $\alpha < 1$? I tried to use some of the possible definition of fractional laplacian, ...
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Singular value decomposition in the language of operator theory

Let $H_i$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H_1,H_2)$ be compact, $|A|:=\sqrt{A^\ast A}$ and $\sigma\in\mathbb R$. How would we describe the singular value decomposition of $A$ in ...
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Spectral decomposition of a linear transformation

I have a linear transformation given by And I need to find its spectral decomposition. I want to represent T as a matrix so I could find it's eigenvalues , and then calculate their eigenspaces or ...
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What is the meaning of eigenvalues in adjacency matrices?

Consider a directed, asymmetric Graph $G$ with its adjacency matrix $A_G$. In the context of recurrent neural networks we use the largest eigenvalue of $A_G$ (i.e., spectral radius $\lambda$) as a ...
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Definition of the Lyapunov exponents for compact operators

There is the following well-known result by Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exponents: Let $H$ be a $\mathbb R$-Hilbert space, $A_n\in\mathfrak L(H)$ be ...
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Eigenvalues of the fractional power of a compact operator

Let $A$ be a compact self-adjoint linear operator on a $\mathbb R$-Hilbert space $H$ and $(\lambda_n)_{n\in\mathbb N}\subseteq\mathbb R$ denote an enumeration of the spectrum $\sigma(A)$ with$^1$ $$|\...
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What can we infer from a growth bound What can we infer from the condition $\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|}n\le0$?

Let $A_n$ be a compact linear operator on a $\mathbb R$-Hilbert space $H$. What can we infer from the condition $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0\tag1$$ and in ...
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Adjacency spectra of a graph interpretation

I'm not a mathematician and I have a question about spectral graph theory. Is it possible to conclude that we have a fully connected network, if an adjacency spectra of a graph is continuous with no ...
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How are eigenvalues and eigenfunctors of operators like Laplacian understood?

Normally you would consider eigenvectors only of operators from the same space to itself. But the Laplacian is usually defined on $C^2$ into $C$, which is a supset of $C^2$. Which leads to my ...
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Which linear operator has a spectrum that corresponds to the physical light spectrum?

If there is only one spectrum for light, which is a continuous spectrum, and Wikipedia says the spectrum that physicists use is the same as the spectrum from math, Which linear operator corresponds to ...
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Spectrum of restriction to invariant subspace

Let $H$ be a separable Hilbert space and let $\mathcal{B}(H)$ denote the algebra of linear bounded operators on $H$. Let $T \in \mathcal{B}(H)$ and let $M$ be a non-trivial closed invariant subspace ...
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Conclude operator convergence from convergence of the eigenspaces

Let $H$ be a $\mathbb R$-Hilbert space, $\Phi,\Phi_n\in\mathfrak L(H)$ be compact and self-adjoint, $I:=\mathbb N\cap[1,\operatorname{rank}\Phi],I_n:=\mathbb N\cap[1,\operatorname{rank}\Phi_n]$ and \...
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64 views

Show that each point in the spectrum of $T$ is an eigenvalue and compute $\|T\|$ [closed]

Let $H$ be a Hilbert space. Let $T \in L(H)$ be a nonzero normal operator such that $T^5 − 2T^3 + 4T = 0.$ Given that, how do we show that each point in the spectrum of $T$ is an eigenvalue? And what ...
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23 views

Point and continuous spectrum of a multiplication operator in $L^2(\mathbb{R})$

Let $M_\alpha f(x) = sin(\frac{x}{\alpha})f(x)$. Compute its spectrum. My idea is to find the measure of the set $A_\lambda = \left\{x \in \mathbb{R} : sin(\frac{x}{\alpha}) = \lambda\right\}$. If ...
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31 views

Property of spectrum

Let $T$ be an operator on $X$ with non-empty resolvent set, and let $\lambda _0 \in \rho (T)$. Show that $\lambda \in \sigma (T)$ if and only if $(\lambda _0 − λ)^{−1} \in \sigma(R(\lambda _0; T))$ ...
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24 views

Norm and eigenvalues of a multiplication operator

Let $M$ be a multiplication operator from $L^2(\mathbb{R}, \mathbb{R})$ to itself, written as the following: $Mf(x)= (-3\chi\left\{x \in [-5,1]\right\} + 2\chi\left\{x \in [-1,2]\right\})f(x)$. ...
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86 views

If $A$ is compact and self-adjoint, then $A=U^\ast DU$ for some orthogonal $U$ and diagonal $D$

Let $H$ be a $\mathbb R$-Hilbert space $A\in\mathfrak L(H)$ be compact and self-adjoint $J:=\mathbb N\cap[0,|\sigma(A)\setminus\{0\}|]$ and $(\mu_j)_{j\in J}$ be an enumeration of $\sigma(A)\setminus\...

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