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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What is the difference between "singular value" and "eigenvalue"?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is "singular value" just another name for eigenvalue?
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Example of a compact set that isn't the spectrum of an operator

This question is a follow-up to this recent question and related to that one. Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset \mathbb{C}$ that can't be the spectrum of a bounded operator…
t.b.
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How to justify solving $f(x+1) + f(x) = g(x)$ using this spectral-like method?

Let's say that I want to find solutions $f\in C(\Bbb R)$ to the equation $$ f(x+1) + f(x) = g(x) $$ for some $g\in C(\Bbb R)$. I can write $f(x+1) = (Tf)(x)$ where $T$ is the right shift operator and rewrite the equation suggestively as $$ (I+…
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Does $\sigma(T) = \{1\}$ and $\|T\| = 1$ imply that $T$ is the identity?

Suppose that $T$ is a bounded linear operator on a complex Banach space X and that we know that $\sigma(T) = \{1\}$ and $\|T\| = 1$ (i.e. the spectrum of the contraction $T$ consists only of a single point, 1). Does it follow that $T$ is the…
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What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find such results? Moreover, is there a counterpart of…
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Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $$Tf(x)=\int_{0}^{x} \!…
yaoxiao
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If $T$ is bounded and $F$ has finite rank, what is the spectrum of $T+F$?

Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always at most countable? Update: If $\sigma(T)=\{0\}$…
Theo
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Symmetries and eigenvalues of the Laplacian.

Lets consider a domain $\Omega \subseteq \mathbb R^2$ smooth enough, and the eigenvalue for the laplacian \begin{align} -\Delta u &= \lambda u &x\in\Omega\\ u &= 0 &x\in \partial \Omega \end{align} I am interested in an explicit relation between…
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Does an unbounded operator $T$ with non-empty spectrum have an unbounded spectrum?

It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their spectrum are empty or the whole complex…
Tomas
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Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension. If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ is injective, and $R(aI-T)$ is not $X$. If…
Strongart
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Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know that $L^*=R$ so these operators are Hilbert-space…
DinkyDoe
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Intuition on spectral theorem

In the last month I studied the spectral theorems and I formally understood them. But I would like some intuition about them. If you didn’t know spectral theorems, how would you come up with the idea that symmetric/normal endomorphisms are the only…
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Can spectrum "specify" an operator?

Given a bounded operator $A$ on a Banach space $X$, one may find the spectrum $\sigma(A)\subset{\bf C}$. Here are my questions: Given some set in the complex plane, say, $S\subset{\bf C}$, can one find an operator $A$ such that $\sigma(A)=S$? Is…
user9464
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How to understand the spectral decomposition geometrically?

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = \sum_{i=1}^k\frac{1}{\lambda_i}e_ie_i'$$ How to understand spectral decomposition and…
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Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex plane. However, every proof I know is by brute…
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