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

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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 ...
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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 ...
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Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio $$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$ is a measure of the "rank-one-ness" ...
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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, $S\subset{\bf C}$, ...
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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 ...
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Find the spectrum of the linear operator $T: \ell^2 \to \ell^2$ defined by $Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$

Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set: $$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$$ for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
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Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
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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 ...
11
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438 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
11
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250 views

why is the spectrum of the schrödinger operator discrete?

let (M,g) be a compact riemannian manifold. Then the spectrum of the Schrödinger opartor $H=-\Delta +V$ with bounded potential V acting on $L^2(M)$ consists of discrete Eigenvalues ...
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Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
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How to understand 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} = ...
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An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
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246 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
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635 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
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144 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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Help computing an integral for Green's function of a discrete Laplacian on a square lattice

I need to calculate the following integral: $$ \int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2 $$ I have tried to use some contour ...
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678 views

Spectral radius inequality

Suppose $A,B \in M(n \times n, \mathbb{C})$ or $ A,B \in M(n \times n, \mathbb{R}) $. Under wich hypothesis can I state that: $\rho(AB) \leq \rho(A)\rho(B)$ ?
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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 $X$, $aI-T$ ...
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What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
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Spectrum of a compact operator

If the spectrum of a compact operator is finite, I don't understand why $0$ has to be a member. I have proved that for all $\epsilon > 0$, there is only a finite number of eigenvectors which have ...
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Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
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Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
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spectrum of the right shift operator

Here is the question: Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$? Here is what I thought: If one wants to prove that the operator ...
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Eigenvalues of doubly stochastic matrices

There was a long standing conjecture stating that the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of regular $k$-gons anchored at $1$ in the unit ...
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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 ...
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Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
7
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Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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Spectral measure of the multiplication operator

I have the following question: let $(X,\mathcal B,\mu)$ be a finite measure space and consider the operator $T\colon L^2(X,\mu)\to L^2(X,\mu)$ given by $Tf(x)=\varphi(x)f(x)$, where $\varphi\colon ...
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Is there a residue theorem for holomorphic operator-valued functions?

I'm wondering whether there is such a thing as a "residue theorem for holomorphic operator-valued functions". More precisely, I want to evaluate an integral of the form $P:=\int_{\Gamma} (A(\lambda) ...
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Convergence of spectra under strong convergence of operators

Say $\left\{A_n\right\}$ is a sequence of bounded self-adjoint operators on a separable Hilbert space, converging in strong operator topology to a (bounded, self-adjoint) operator $A$. Denote the ...
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Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
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Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
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Do elliptic operators on Riemannian manifolds have a regularizing effect?

I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential ...
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Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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Spectral decomposition of a normal matrix

I'd like to find the spectral decomposition of $A$: $$A = \begin{pmatrix} 2-i & -1 & 0\\ -1 & 1-i & 1\\ 0 & 1 & 2-i \end{pmatrix}$$ i.e. $A=\sum_{i}\lambda_i P_i$ where ...
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About some properties of the heat kernel

Maybe my questions are stupit, but I'm so unsure about some properties of the heat equation. Suppose we have a compact and smooth manifold M (possibly with boundary), then in a lot of books they ...
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A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
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Doubt about integration over a curve

I'm reading a paper and I have a doubt: Suppose we have a compact Riemannian manifold and an eigenfunction of the Laplace operator $\Delta u=-\lambda u$. Suppose in addition that we have the following ...
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renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
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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 ...
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Explicit expression for eigenpairs of Laplace-Beltrami operator

In $R^n$, the Laplace-Beltrami operator is just the Laplacian, and its eigenstructure is well known. There are also explicit expressions for the eigenvalues/eigenvectors of the Laplace-Beltrami ...
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What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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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 ...
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spectrum of the “discrete Laplacian operator”

In numerical analysis, the discrete Laplacian operator $\triangle$ on $\ell^2({\bf Z})$ can be written in terms of the shift operator $\triangle=S+S^*-2I$ where $S$ is the right shift operator. ...
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Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
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What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...