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

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Spectrum of an element in sub-algebra: $\sigma_A(b)\setminus \{0\}\subseteq \sigma_B(b) \setminus \{0\}$

Please help me to prove this:(or give me some references for this.) Thanks very much! Let $A$ be a (unital) algebra and $B\subset A$ a (unital) sub-algebra. Then for all $b\in B$: ...
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0answers
213 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
3
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1answer
733 views

are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
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2answers
366 views

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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1answer
140 views

The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$? Edit: What if the matrix has positive ...
3
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1answer
166 views

Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign

I want to prove (or disprove) the following statement: If $A$ is a square matrix with non-negative elements that has spectral radius less then $1$, then any matrix obtained from $A$ by ...
4
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0answers
259 views

Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$

Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum. To approach ...
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286 views

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 ...
2
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1answer
163 views

Spectral radius and powers of a $2\times 2$ block matrix

The problem I'm struggling is the following: Let $n$ be a positive integer and let $A=% \begin{pmatrix} B & C\\ C & B \end{pmatrix} \in\mathcal{M}_{2n}(\mathbb{R}_{+})$, where ...
1
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1answer
130 views

Spectrum Of The Laplacian Question

I'm given the following question (from Davies' book - Spectral theory of differential operators): Use the theorem (*) below to prove that if $\Omega$ is a convex region in $ \mathbb{R}^2 $ , then: $ ...
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2answers
152 views

Symbol Of Differential Operators

When we are given a differential operator of the form $Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $ , we can define the symbol associated with it to be the function: $a(x,y) := \sum_\alpha ...
3
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0answers
89 views

Spectrum of a bounded linear operator [duplicate]

Possible Duplicate: Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $ Let $A$ be a bounded linear operator on a (complex) Hilbert space. I want to prove that ...
11
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1answer
362 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 ...
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1answer
120 views

Integral with spectral decomposition

Let $A:H\longrightarrow H$ be a self-adjoint operator, where H is an Hilbert space. Let $(E_{\lambda})_{\lambda}$ be the spectral decomposition of $A$ and $\lambda_0$ a regular value of A with finite ...
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1answer
133 views

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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1answer
291 views

Is it true that the Laplace-Beltrami operator on the sphere has compact resolvents?

We consider the Riemannian structure on the sphere $\mathbb{S}^n$ seen as a submanifold of $\mathbb{R}^{n+1}$ and the Laplace-Beltrami operator defined on $C^\infty(\mathbb{S}^n)$ by the equation ...
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1answer
182 views

Fourier Transform of a Covariance Function for Spectral Simulation

I am learning about generating Gaussian random fields by spectral simulation... If I have a covariance function $C(h)$, then the spectral density is the Fourier transform of $C(h)$: ...
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0answers
225 views

Orthogonal projection and normal operators

Let $G$ be normal operator with compact resolvent such that $\ker G$ is different from $\{0\}$. Now Let $P$ be the orthogonal projection onto $\ker G$ and consider $G' = G + P$. Please, I want an ...
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1answer
48 views

Well definededness of integration with respect to a projection valued measure

Let $(X,\mathcal{F})$ be a measurable space and let $E:\mathcal{F}\to\mathscr{B(H)}$ be a spectral measure. Let $\phi\in B(X)$ be a simple function whose image is ...
2
votes
1answer
101 views

Family of Self-Adjoint Operators that are Multiplications on a Common $L^2(\mu)$?

Suppose that $H$ is some (complex) Hilbert space and that $\{T_\alpha: \alpha \in I\}$ is some collection of bounded self-adjoint operators on $H$. A version of the spectral theorem states that for ...
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2answers
271 views

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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0answers
278 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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2answers
2k views

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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1answer
251 views

Approximate Point Spectrum is subset of Spectrum

I'm trying to prove that if $\lambda$ an approximate eigenvalue of $T$ then $\lambda \in \sigma(T)$, but I can't work out how to do it. Could someone give me a hint, or point me in the direction of a ...
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0answers
160 views

Using Rayleigh Quotient to approximate the first eigenvalue of the Laplace operator on the unit disk

Let $D\subset\mathbb{R}^{2}$ unit disk, the first eigenvalue of the Laplace operator holds: $\lambda_{1}=\inf\left\{ \frac{\int_{D}\left|\triangledown ...
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1answer
284 views

Compact operator? self adjoint operator? Stirling's formula

Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $M\colon\ell^2 \to\ell^2$ as follows: $$S(x_1,x_2,x_3,x_4,\ldots)=(0,x_1,x_2,x_3,\ldots) $$ and ...
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1answer
550 views

Eigenvalues of the 1D laplacian with mixed boundary conditions

I am trying to find the eigenvalues and eigenvectors of the Laplacian with mixed boundary conditions on $[0,L]$: More precisely: $$X''(x) = \lambda X(x)$$ with $X'(0)=0$ and $X(L)=a$. I know how to ...
4
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1answer
227 views

“Algebraic multiplicity” for eigenvalues of a Sturm-Liouville-like problem?

Following Coddington-Levinson's book Theory of ordinary differential equations, chapter 7: "Self-adjoint problems on finite intervals", let us consider the eigenvalue problem $$\pi(l):\begin{cases} ...
5
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2answers
207 views

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is ...
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2answers
370 views

Approximate eigenvalue and continuous spectrum

Let $\mathcal{H}$ be a Hilbert space and let $A: \mathcal{H} \rightarrow \mathcal{H}$ be a bounded operator. While studying different definitions of the continuous spectrum of $A$ (one using ...
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1answer
266 views

How to solve X*A=C matrix equation where two (X and A) matrices are unknown?

I have a spectroscopy problem that boils down to a matrix equation where X*A=C. I take N observations each consisting of 3 detector readings and my detectors suffer from some amount of cross-talk ...
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1answer
295 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
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1answer
1k views

How to find eigenfunctions of a linear operator (follow-up question)

This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ...
3
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1answer
453 views

How to find eigenfunctions of a linear operator

I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral. As a special case suppose I am interested in this function: ...
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3answers
860 views

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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1answer
96 views

Characterizations of the form domain for unbounded selfadjoint operators

This question follows from this one and especially from Willie Wong's answer: link. In Reed & Simon's book Methods of modern mathematical physics, vol. I, pag.277, the form domain of a ...
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102 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
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4answers
14k views

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 ...
3
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1answer
241 views

Spectral radius, and a curious equality.

Given a $N\times N$ matrix $A$ over $\mathbb R$. Let $ \rho\left( A \right) = \max \left\{ {\left| \lambda \right|;\lambda \mbox{ eigenvalue of }A} \right\}$. Someone told me that, the following ...
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1answer
167 views

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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0answers
335 views

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal That says it all, but let me be more specific. (By the way Wikipedia says this: "The product of normal operators that commute ...
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0answers
84 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
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219 views

A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)? I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
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1answer
644 views

How to generate noise signal?

What is the simplest formula of some noise signal? $A(t)=...$ where t is time. What is the name of a noise, which power spectral density is gaussian? EDIT 1 Actually I need a function which can ...
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1answer
111 views

Unitary equivalence to multiplication by x of the sum of a shift and its adjoint

We define $H=l^2(\mathbb{Z})$, $S\in L(H)$ to be the left (bilateral) shift and we look at $T=S+S^*$ ($S^*$ is actually the right shift). We need to prove that the spectrum of $T$ is $[-2, 2]$ and ...
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0answers
111 views

relation between inner product and spectrum

There is a question that puzzles me, so may be someone here has an answer. Assume we have a symmetric operator $A$ that is defined on a space $D$ that is dense in $L^2$, so $A:D\rightarrow L^2$, and ...
4
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3answers
585 views

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 ...
2
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0answers
98 views

Bounds for eigenvalues, perturbation theory

Consider $-\Delta$ defined in $H^2(\Omega)\cap H_0^1(\Omega)$, $\Omega$ a smooth bounded domain of $\mathbb{R}^n$. Let $g\in L^{\infty}(\Omega)$, $a\leq g(x)\leq b$. Show that, if ...
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1answer
83 views

change of spectrum under diagonalisation

I have the following question. Let $T\colon \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on a Hilbert-space $\mathcal{H}$. By spectral theorem we know that there exists a measure ...
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1answer
495 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 ...