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

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The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
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Adjacency matrix of directed graph

I am given adjacency matrix $A$ of directed graph. $A(x,y)$ counts the number of edges from $x$ to $y$. I want to show that if $A$ has constant outdegree $d$: (i) For any eigenvalue $\lambda$, we ...
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232 views

How to use Parseval' identity( Plancherel)? [duplicate]

(May be this is very basic question for MO) Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix ...
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Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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when the equality $\sigma_A(a)=\sigma_B(\phi(a))$ hold?

Let $A$ and $B$ be unital commutative C*-algebras and let $\phi : A\to B$ be a *-homomorphism with $\phi(1_A)=1_B$. Under what conditions the equality $\sigma_A(a)=\sigma_B(\phi(a))$ holds where ...
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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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29 views

A couple of proofs on a spectrum

Let $T$ be a normal bounded operator. Let ${\lambda}$ be in $({\sigma}(T))$. Without invoking general algebra theories, show that: a) $p({\lambda},{\lambda}^*)$ is in $({\sigma}(T))$ for all ...
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Spectrum of perturbed operator's

Let $G$ be a normal operator with compact resolvent acting on a Hilbert space $H$ such that $\ker G \neq \{0\}$. If $P$ denotes the orthogonal projection onto $\ker G$, and if $\{\lambda_n\}$ are ...
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If $\lambda$ is isolated in $\sigma(u)$, then $E(\left\{\lambda\right\})(H)=\ker(u-\lambda)$.

This is a Question 2.11 from Murphy's book: C$^*$-algebras and Operator Theory: Let $H$ be a Hilbert space. Let $u\in B(H)$ be a normal operator with spectral resolution of the identity $E$. (a) ...
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32 views

(Girko's) circular law beyond the standard normal distribution

I found this circular law , which states a uniform distribution of eigenvalues in the unit circle of the complex plane when the eigenvalues are of a asymmetric random matrix of which the elements are ...
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69 views

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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16 views

characterisation up to unitary equivalence

My book says that the spectral theorem for compact normal operators characterises compact normal operators up to unitary equivalence. It doesn't expand on this so I was wondering what does this mean ...
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25 views

The eigenspace of 0 problem

In the spectral theorem for a compact normal operator, do we exclude the eigenspace corresponding to 0 (assuming its an eigenvalue) from the space decomposition. My reason for asking is this: Every ...
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40 views

properties of norms of operators

Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum such that $L^{-1}$ is bounded and $$\|Tf\| \leq \Phi_{\eta}(f) + \eta ...
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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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101 views

Direct sum of eigenspaces of a compact operator has finite codimension

In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks.
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operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
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59 views

Spectral decomposition

For a compact normal operator, the space can be written as the sum of generalized eigenspaces. So every element can be written as a linear combination of the eigenvectors, one from each eigenspace. ...
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54 views

Spectral decomposition for normal compact operator

My book says $Tx=(\alpha x_{\alpha})$ where the $\alpha$ are eigenvalues the of T. But the image of an operator is not in general a sequence. Do they mean these are the scalars in the linear ...
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pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
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Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
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Clarification about Arveson exercise

In an exercise at the start of the book of Arveson "A short course on Spectral Theory"(page 5 exercise 5), while asking to prove that the operator $Kf(x)=\int_{0}^{1}k(x,y)f(y)dy$, (where k is a ...
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Projections in spectral decomposition.

In my quantum mechanics book the spectral decomposition of operator $A$ is given as $A=\sum\limits_j\lambda_jP_j$ where $\lambda_j$ are the eigenvalues of matrix $A$ and $P_j$ is the orthogonal ...
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42 views

Resolvent's estimation.

Let $H$ be a Hilbert space and $L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I would ask about this inequality because I don't understand it$$\displaystyle{\|(L- \lambda ...
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96 views

Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
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49 views

Perturbation of a matrix with negative eigenvalues

Let $A$ be a square matrix with all eigenvalues negative. What is the relationship between the $\lambda_\max$ of perturbed matrix $A + X$ and the norm of the perturbation $\|X\|$? PS: I know that the ...
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86 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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21 views

Spectral Relaxation and Eigendecomposition

Suppose I have the following optimization problem: $\underset{\mathbf{x}}{\mathop{\max }}\,{{\mathbf{x}}^{T}}A{{\mathbf{x}}^{T}}$ s.t $\mathbf{x}\in {{\left\{ -1,1 \right\}}^{N}}$. One possible ...
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Fast Gauss-Seidel convergence on low rank matrices

I stumbled upon the following remarkable fact when experimenting with the Gauss-Seidel iterative solver: First I construct a low-rank symmetric positive semi-definite matrix $A = M^TM$ with M a ...
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How to show that the spectral radius of a binary tree approaches exp(1) as the N tends to infinity?

How can I prove mathematically that the spectral radius of a binary tree approaches e as the number of nodes tends to infinity? I mean it is true that the increase in nodes number is exponential but ...
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Characterization of all matrices with unit spectral radius under constraint

Let $A \in \mathbb{R}^{n \times n}_{\geq 0}$ be a symmetric matrix with positive row sums $\mathbf{d} := A\mathbf{1} > 0$. I am interested in characterizing all those positive diagonal matrices $Z ...
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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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Spectrum of product from two selfadjoint matricies

If I have 2 selfadjoint matricies A,B given, is the spectrum of $A B$ always real? I know that $A B$ is not necessary a selfadjoint matrix, but are some properties of the spectrum preserved?
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A transformation recipe for functional calculus of a self-adjoint operator?

Consider a self-adjoint operator $\operatorname{A}$ on a Hilbertspace $\mathcal{A}$ and its spectral decomposition according to the spectral theorem: $$A = \int_{\mathbb{R}} \lambda \;dP_\lambda$$ ...
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Spectrum of the Orr Sommerfeld equation

The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions ...
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Characterization of central (modulo the radical) elements of a Banach algebra

Let $A$ be a Banach algebra and $Z(A)=\{a \in A:ax-xa \in $ Rad $ A \ \forall x \in A\}$ be the centre (modulo Rad$A$). Then TFAE: 1) $a \in Z(A)$ 2) $\exists M >0$ such that $\rho(a+x)\leq ...
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Resolvent R(1) of the Laplace operator not compact

I want to show that $$R_\Delta(1):=(1-\Delta)^{-1} $$ is not compact in $\mathbb{R}^3$. I have found that for $\chi_{B}$ being the characteristic function for a set $B\subset\mathbb{R}^n$, ...
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262 views

Spectrum of operator

Like my previous question, I'm considering the same space and operator: Hilbertspace adjoint But this time I am trying to determine the spectrum of $T$. I feel like I'm messing up my definitions a ...
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invariant inner product on eigenspace

I have several questions about the following corollary: "Let G/H be a riemannian homogeneous space where G is a compact Lie group. Let $E_{\lambda}=\lbrace f\in C^{\infty}(G/H) : -\Delta f= \lambda ...
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1answer
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regarding a proof of $\|\theta(e^{i\lambda})\|^2$

When studying the spectral representation of time series, I read the following formula, I am not clear how to prove the second equation. I expand the left side of the second equation with the ...
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1answer
31 views

spectum of self adjoint operators

Let $H$ be an Hilbert space and $S = \displaystyle{ \sum_{i=1}^nS_i}$ where $S_i$ (i=1...n) is self adjoint with compact resolvent . is it true that the spectrum of $S$ is the sum of the spectra of ...
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relations between two linear operators

Let $\alpha,\beta$ be linear operators on a finite dimensional vector space $V$ over field $F$. Let $\gamma=\alpha\circ\beta$ and $\delta=\beta\circ\alpha$. Prove that: (1). $m_\delta(x)$ divides ...
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Eigenvalues and eigenvectors of AB and BA in Endomorphisms [duplicate]

Let A and B endomorphisms of a finite vector space over a field F , Prove or a counterexample to the following statements: a) all eigenvector of AB is an eigenvector of BA; b) all eigenvalue of AB is ...
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Limit of the spectrum in Banach algebra

Let $A$ an unital complex Banach algebra, $a_{n} $ is a sequence such that $\lim_{n\to \infty}a_{n}=a$. What is the relation between $\lim_{n\to \infty}\sigma(a_{n})$ and $\sigma(a)$. I think that it ...
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81 views

Multiplication operator on Hilbert space

i looked to the question Spectrum and point spectrum of this operator. I will go further with asking. We know that $T$ is well-defined iff $(\lambda_n)\in\ell^{\infty}$. But if ...
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1answer
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Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal.

The question is: Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal. Then I have to find the spectral decomposition of $T^{-1}$. At first I tried to prove it by ...
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301 views

Singular value decomposition of an arbitrary anti-symmetric ($A=-A^{T}$) complex matrix

I am a physicist and very much used to the fact that any self-adjoint matrix ($H^{\dagger} =H$) in a finite-dimensional complex linear space can be uniquely specified by (a) the set of its (real) ...
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1answer
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$T\in B(H)$ normal and left invertible implies $T$ invertible?

My question is what's written in the title, that is, if $T$ is a normal operator on a Hilbert space $H$, and $T$ is left invertible, is it necessarily true that $T$ is invertible? Actually, the more ...
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Why is this spectrum the closure?

I've started to learn about spectral theory and I'm looking at some examples. The spectrum is defined as $$ \sigma(a) = \{\lambda \in \mathbb C | a-1\lambda \text{ is not invertible} \}$$ If $A= ...