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Questions tagged [spectral-theory]

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

Spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

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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 (r,\varphi,\vartheta)}_{\llap{ \text{wave function in spherical…
71GA
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Dirichlet eigenvalues of Laplace-Beltrami operator on subsets of the sphere

I am interested in the Dirichlet eigenvalues of $-\Delta_{\mathbb S^n}$ on open subsets $D \subsetneq \mathbb S^n$. If $D$ is a hemisphere, then the first eigenvalue is equal to $n$, with the eigenfunction given by $x_1^+$ for some choice of…
anon
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Eigenvalues of the "Laplacian" on [0,2$\pi]\subset\mathbb{R}$

for the dirichlet eigenvalue Problem on compact and connected Riemannian manifolds, the eigenvalues of the laplacian consists of a discrete sequence. On the other hand, if we consider $[0,2\pi]\subset\mathbb{R}$, the Eigenvalues of the laplacian…
Braten
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Spectrum of left shift operator $L\in B(H)$

Let $H$ be a Hilbert space with an orthonormal base $e_i$ and $L$ the left shift operator $L\in B(H)$: $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$. I computed the spectrum could someone please tell me if this is right? My work: $\lambda \in \sigma…
user167889
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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 two-torus, otherwise known as a triangular…
Booger Rooger
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Spectral radius inequality for normal matrices.

I've read in several places now that $\rho(AB) \leq \rho(A)\rho(B)$ holds if the matrices are normal. I've scoured the web for a proof since I can't seem to get there myself. How would one go about to prove this statement? Links to proof is also…
CupinaCoffee
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Examples for spectrum of an operator

Looking for easy-to-understand examples for the spectrum of an operator, preferably so that they exposed some special properties. The right shift is a nice example of an operator which does not have a point spectrum. What is the simple most example…
Anno2001
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use the spectral decomposition to show that $x^T A x \le \lambda _1 ||x||^2$

use the spectral decomposition to show that $x^T A x \le \lambda _1 ||x||^2$ The spectral decomposition says that $A = \sum \lambda _i v_i v_i ^T$ Multiplying both sides by $x^T$ and $x$, we get $x^T A x = x^T \sum \lambda _i v_i v_i ^T x$ Where do…
ssss
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EDO by separation of variable - When are we certain that $\lambda \geq 0$?

If we use the separation of variables to solve $u_t-u_{xx}=-u$, $u=u(x,t)$, then we obtain $f(x)g'(t)-f''(x)g(t)=-f(x)g(t) \iff \frac{g'(t)}{g(t)}+1=\frac{f''(x)}{f(x)}$. The only way that a function of $x$ can equal a function of $t$ is for both…
user316765
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The topology of the spectrum of a linear operator

In general a spectrum of a linear operator has a decomposition into three parts: point spectrum, continuous spectrum and residual spectrum. What I'm interested in is the topology of these parts of spectrum. Is the point spectrum always discrete?…
anonymous67
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spectrum of operators

Let $X$ be a Banach space and $T: X \rightarrow X$ is a bounded operator. The spectrum of $T$ is defined by $$\sigma(T) = \{\lambda \in \mathbb{C}; \lambda I - T ~~ \mbox{is not invertible}\}$$ It is well known that $\sigma(T)$ is a bounded…
salma
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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 $ N(\lambda)= \sum_{n=0}^{\infty}H( \lambda - \lambda _{n})…
Jose Garcia
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Spectrum of operator based on orthogonal vectors

I need to find the eigenvalues and vector for the operator in hilbert space $H$: $$Tx=\langle x,v\rangle w+\langle x,w\rangle v,$$ where $\langle v,w \rangle =0$, but not necessarily with norm=1. I already checked that it is compact and self adjoint…
Benuci
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Proving $\sigma(f(A))=\overline{f(\sigma(A))}$ with the spectral theorem

I have to show that for a self-adjoint operator $(A,D(A))$ on a seperable Hilbert space $\mathcal{H}$ and $f\colon \mathbb{R}\to \mathbb{R}$ continuous and bounded that \begin{align*} \sigma(f(A))=\overline{f(\sigma(A))}\, . \end{align*} I thought…
putti.123
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Spectral representation of a matrix

I am given an operator $T: \mathbb{C}^3 \rightarrow \mathbb{C}^3$ that is represented by the matrix $$ \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$ Now I am supposed to find the spectral family and the resulting spectral…
Lauchmelder
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