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

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Article in spectral theory

Can someone please help me to understand how we prove the first inequality (Page 7) in this article http://arxiv.org/pdf/1510.01567.pdf It seems that we use the Young inequality but i didn't know ...
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30 views

Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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Two formula for an operator

we define the operator P with domain $C_0^\infty(R^{2n})$ by $P=y.\partial _x -\partial _x V(x).\partial_y-\Delta_y+\frac{|y|^2}{4}-\frac{n}{2}$ I want to prove that $P=X_0 +\sum _{i=1}{^n}X_j^*X_j$ ...
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About the Mercer's theorem.

Doesn't the the Mercer's theorem say something stronger than just the spectral theory of compact self-adjoint operators on a Hilbert space applied to the reproducing "kernel" function? As in if I ...
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15 views

Inequality for an operator

We consider the operator $P$ with domain $C_0^\infty(R^{2n})$ defined by $P=y.\partial _x -\partial _x V(x).\partial_y-\Delta_y+\frac{|y|^2}{4}-\frac{n}{2}=X_0 +\sum _{i=1}{^n}X_j^*X_j$ where $X_0=y.\...
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Problem with Spectral Theorem Proof

Claim: Let A $\in \mathbb{R}^{n \times n }$ be symmetric. Then there is an orthonormal basis of $\mathbb{R}^n$ consisting of eigenvectors of A. Sketch of proof: Induction on n. Claim is clear for ...
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44 views

The density of $C^1[0,2\pi]$

I am not sure if the inclusion $\{f \in AC[0,2\pi]: f(0)=f(2\pi)=0\}\subseteq \overline{\{f \in C^1[0,2\pi]: f(0)=f(2\pi)=0\}}.$ Here $C^1[0,2\pi]$ is the set of continuously differentiable functions ...
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30 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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19 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
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33 views

L2 Norm: Unfamiliar notation

In this article that I am reading, I am given a non-negative spectral function $w(\lambda)$ which is "interpreted as a weight function determining the scalar product of two functions $f(\lambda)$ and $...
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27 views

Spectral measure of a stationary time series

Let $(Z_t)$ be white noise with $E[Z_t^2]=1$ and $A$ and $B$ random variables such that $E[A] = E[B] = 0$, $E[A^2] = E[B^2] = 1$, $A$, $B$ and the infinite sequence $(Z_t)$ are independent ($(Z_t)$ ...
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Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by $...
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Spectral density of a filtered stationary process

I have a stationary time series $(X_t)$ with spectral density $f_X$, i.e. $$f_X(\lambda) = \frac{1}{2\pi}\sum_{h\in\mathbb{Z}} e^{-ih\lambda}\gamma_X(h)$$ where $\lambda \in (-\pi,\pi]$ and $\gamma_X(...
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If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ergodic....
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Change of variables of a polynomial

Can someone help me please to solve this problem: we consider $V(Y)$ a polynomial in $ R[Y_1,Y_2,..,Y_d] $ I want to prove that there existe an affine change of variables $ Y=AX+B,X=(X_1,X_2,..,X_d)$ ...
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50 views

Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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If spectral radius $\rho(A)<1$ , does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-||A||_{2})$ hold true?

If spectral radius $\rho(A)<1$, does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-\||A||_{2})$ hold true? If it is correct can somebody give me link to the proof for this inequality?
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67 views

ODE system with lower triangular coefficient matrix

Let $v_k$ satisfies the following equation $\frac{d}{dt} v_k(t)=\sum_{i=0}^{k-1}v_k(t)$ with initial data $v_0(t=0)=1$ and $v_j(t=0)=0$. Notice that the coefficient matrix $A$ of this ODE system (...
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Formula connecting the resolvent opeartor andthe spectral density?

I want to know if it is a formula connecting the resolvent opeartor $(\lambda - T)^{-1}$ for a selft-adjoint operator $T$ and its spectral density $e_{\lambda}$. Thank you in advance
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Formula connecting the resolvent and the heat kernels

Using the well known formula connecting the resolvent and the heat operators associated to a selft-adjoint opeartor $A$ \begin{align} (\zeta - A)^{-1} = \int_{0}^{\infty} e^{-\xi t} \, e^{t A} dt; \...
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30 views

Direct Integral: Scalars

Given a Borel space $\Omega$. Regard the Hilbert Space: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mathcal{H}:=\mathcal{L}^2(\Omega;\mu)$$ Denote the Borel Projections: $$E:\mathcal{B}...
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27 views

Largest element in inverse of a positive definite symmetric matrix.

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
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spectrum of unbounded self-adjoint operators

I'm self-studying Lax's functional analysis, and I'm stuck in the chapter introducing spectral theory for unbounded self-adjoint operators. In his book, Lax proved the spectral theorem of this ...
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square root of positive elements preserve order

Let $A$ be a $C^*$ algebra. Show that if $0 \le a \le b$ then $\sqrt a \le \sqrt b$. I've shown that this is true in case $b$ is invertible, here is my proof: $$\|a^{1/2}b^{-1/2}\|^2 = \|(a^{1/2}b^{-...
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80 views

Path - Geometry [closed]

I am currently completing the end of a Bachelor degree in pure mathematics. I would like to work on an interesting project (by myself) this summer in the field of spectral geometry. Does someone could ...
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37 views

Spectral Measures: Poisson

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}H\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad H=\int\...
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29 views

Pseudo-resolvent function

Let $\emptyset \neq D$ a open set in $\mathbb{C}$ and $J: D \to B(E)$ a continuos function such that $J(\lambda) - J(\mu) = (\mu - \lambda)J(\lambda) J(\mu)$ where $E$ is Banach space. We must show ...
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Compactness of the resolvent

I want to prove the following proposition: owning to the $ H^2_{loc}(\mathbb{R}^n)$ regularity when solving $\Delta^{(0)}_Vu=f\in L^2$, the compactness of the resolvent of $\Delta^{(0)}_V$ is a ...
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41 views

Bound self-adjoint operator

Assume we have a positive (so that we can take the square-root by functional calculus) self-adjoint operator $H: D(H) \subset \mathcal{H} \rightarrow \mathcal{H},$ then we can define $V:=D(H^{\frac{1}{...
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15 views

Minimum of the Spectrum for a Closed Operator

If $T$ is closed is it true that if $\lambda=\min\{\sigma(T)\}$ and $((T-\lambda)f,f))=0$ then $(T-\lambda)f=0$? This is for a fixed $f\in\mathscr{H}$.
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Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: $$\varphi\geq0\implies\|\...
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35 views

Show that $||(kI-T)^{-1}|| \le \frac{1}{d}$

Suppose that $T \in BL(H)$ where $H$ is a Hilbert Space. Let$k \in \mathbb{C}$. Let $d=distance(k,W(T)) \gt 0$. $W(T)=\{\lambda \in \mathbb{C}: \lambda=<Tx,x>, ||x||=1, x \in H\}$. Show that $||(...
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Expansion of an eigenfunction $\psi$ into a Fourier series

I was going through a paper by Jean Bourgain and it states that an eigenfunction $\psi$ of the Laplacian $\Delta$ with eigenvalue $-\lambda$ on the flat $n-$torus $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^...
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37 views

Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using, H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\...
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1answer
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irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when $...
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33 views

Maximal Accretive Implies Injective

I'm having trouble proving that $B $ is essentielly maximal accretive implies that there existes $ a>0 $ such that $(A^∗+a Id)$ is injective. where B is the closure of the operator A and A* is ...
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28 views

Equality of two measures

I am trying to prove the following statement. Let $F$ and $G$ be two finite measures on $((-\pi,\pi],\mathcal{B}((-\pi,\pi]))$ such that $$\int_{(-\pi,\pi]}e^{ih\lambda}\,dF(\lambda)=\int_{(-\pi,\pi]}...
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26 views

Fokker Planck operator is accretive

I would like to show that the Fokker Planck operator with domain $C_0^\infty \left(R^{2n}\right)$ defined by $$K= v.∂_x − (∂_xV (x)).∂_v + (−∂_v +v/2).(∂_v +v/2)$$ is essentially maximal accretive ...
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Uniqueness of positive square root of postive element in C* algebra

If a is a positive element then it has a unique positive square root, i.e. a unique b positive such that b^2=a. I understand the existence part of the proof. If ...
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Does the word “spectrum” in linear algebra have different meanings?

I'm reading several papers that refer to the spectrum as the set of all possible eigenvalues of a matrix, i.e., counting multiplicity, so that a list such as $\sigma = (\alpha_1, ... \alpha_n, 0, 0, 0,...
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Spectrum is finite or unbounded

Let $X$ be a Banach space and $A:D(A)\subset X\to X$ be a linear and closed operator with $\rho(A)\neq \emptyset$. Suppose that the map $$j:\left(D(A),\|\bullet\|_A\right)\hookrightarrow \left(X,\|\...
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Determining eigenvalues of a differential or integral operator in Matlab?

Say I have a differential operator such as $L[\phi] = \frac{\partial \phi}{\partial x}$, or $L = \Delta \phi$, or an integral operator such as $L[\phi](x) = \int_{\partial D} \log(x - y) \phi(y) d\...
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50 views

Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain $D(A)=\{f\...
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Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
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51 views

Proving consequences of spectral decomposition of normal operator

$T$ is a normal operator on finite-dimension complex inner product space $V$. How do I use the spectral decomposition $T=\lambda_1T_1+\cdots+\lambda_kT_k$ to show: a) If $T^n=0$ for some n, ...
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1answer
35 views

Exponential of a self-adjoint operator

Let $\mathcal{H}$ be an Hilbert space. Firstly, I shall define some notions as their definitions may vary: A spectral resolution is a function $E:\mathbb{R}\to\mathcal{L}(\mathcal{H})$ (the space ...
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39 views

Under what conditions is the resolvent set of a linear operator connected?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$. As usual, we define ...
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29 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
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43 views

Inverse spectrum problem - showing the existence of a 2x2 doubly stochastic matrix,

I am working through a couple of problems in Henryk Minc's book, Nonnegative Matrices, as a warm-up to understanding the inverse spectrum problem. This is Exercise 18 of Chapter VII of his book: ...