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

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If $\Delta_{\mathbb{R}^n}u = \lambda u$ and $u(x)=u(x+y)$, then $u_{y}(x):=e^{2 \pi i <x,y>}$

Related to Analysis on Manifolds via the Laplacian page $52$, I would like someone explain to me why if we have a function $u$ such that $\Delta_{\mathbb{R}^n}u = \lambda u$ and $u(x)=u(x+y)$ with $y ...
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$\lim_{k \to \infty} \frac{N(\lambda_k)(2 \pi)^3}{\omega_3 \text{Vol}(\mathbb{S}^2) \lambda^{\frac{3}{2}} } =1 ~\not = 0$

I would like to use Weyl's to confirm my result from the spectrum of $\mathbb{S^2}$. So far I found that the spectrum of $\mathbb{S^2}$ is $\{k(k+1) : k \in \mathbb{N} \cup \{0\} \}$ and each ...
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1answer
45 views

A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
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26 views

Spectrum of an isometric operator [closed]

spectrumIf X is a Banach space and $T: X \rightarrow X$ is an isometry, then either $\sigma (T) \subseteq unit\ circle$ or $\sigma(T)=clD$
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Min-Max principle - Subspaces over $H_0^1$

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem $$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\...
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2answers
272 views

Min-Max Principle with matrices - Understanding $\lambda_2$

Related to the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I am trying to do the same with $$A= \begin{bmatrix} 2 & 0 & 0\\ 0 &...
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27 views

Extending a unitary operator

Suppose that $\mathcal{H}_1$ and $\mathcal{H}_2$ are two separable Hilbert spaces and that $X\subset \mathcal{H}_1$ is a dense subspace (i.e. $\overline{X}=\mathcal{H}_1$). If $\operatorname{W}:X \to \...
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1answer
166 views

$\displaystyle\frac{\langle A x_{\min}, x_{\min}\rangle }{\langle x_{\min}, x_{\min}\rangle }=\lambda_1 \not = \lambda_2$? - Explanations

Related the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I tried to complete the details of mickep's answer, but I have a little problem....
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Complex dispersion relation - example

I am dealing with eigenvalue problems of linear operators $L$ on the real line that can be written as a linear nonautonomous ODE $$ v_{\xi}=A(\xi;\lambda)v,~~\xi\in\mathbb{R},\lambda\in\mathbb{C}, v\...
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1answer
35 views

operator inequality using spectral theorem

Given two densely defined unbounded self-adjoint strictly positive operator $A$ and $L$ in Hilbert space $H$ with domain $D(A) \subset D(L)$ and $\|Lx\| \leq \|Ax\|$ for all $x\in D(A)$, why do we ...
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23 views

Compactness of the resolvant of a Schrodinger operator

I want to prove that the closure of an operator $A$ with domain $D(A)= \mathcal{C}^{\infty}_c(\mathbb{R}^n)$ given by: $A=\Delta+\frac{1}{4}||\nabla V(x)||^2-\frac{1}{2}\Delta V(x)$ has a compact ...
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1answer
8 views

A restriction $A_2$ of a compact self-adjoint compact linear operator $A$ is also compact and self-adjoint?

Let $X$ be an inner product space and let $A$ be a compact and self-adjoint linear operator. Let $p_1$ be an eigenvector of $A$. Let $A_2$ be the restriction of $A$ to $X_2$ where $X_2$ is given by $$...
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1answer
26 views

Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
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1answer
23 views

Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
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Ineqality involving operators

I need your help to solve the following problem: We define the operatorss $a$ and $b$ with domain $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$ by: $a=\partial_x+\frac{1}{2}\partial_xV(x)$ (where $V$ in ...
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47 views

Neumann vs Dirichlet eigenvalue problem - Intuition

What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, ...
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1answer
187 views

Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations

In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ...
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2answers
22 views

Question about Langevin equation

The Langevin equation is given by: $dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$ I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.
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Domain Monotonicity - Neumann eigenvalue problem (Edit)

Related to the question : http://mathoverflow.net/questions/242136/why-m-1-subset-m-2-not-rightarrow-n-m-1-lambda-leq-n-m-2-lambda The Neumann eigenvalues of the rectangle with sides $a$ and $b$ are $...
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1answer
47 views

Spectrum of $T\in \mathcal{L}(E)$, such that $T^n=I$

Let $T:E \to E$ be a bounded linear operator, $E$ infinite dimensional Banach space, such that $T^n =I$, for $n\ge2.$ Show that $\sigma(T)\subset\{-1,1\}.$ My idea is show that $\|T\|=1$ initially,...
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Is this a general identity for the Resolvent? [solved: integral representation of the resolvent]

And if so, what is it called? $$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$ as in Reed-Simon XIII.7 example 1. It is stated there for $H=-i\...
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Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
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1answer
21 views

Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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35 views

Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
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25 views

Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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64 views

How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law

I am trying to prove the Weyl's asymptotic law for eigenvalues. In the document Weyl's law of p. $4$, I have managed to go up to the step $$\tilde{\nu_k} \leq \nu_k \leq \mu_k \leq \tilde{\mu}_k \...
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Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
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$\forall f \in H^k$ is homogeneous of degree $k$, $f=r^k \hat{f}$, where $\hat{f}$ is in function of $\theta$ and $\phi$

In the document Spectral Geometry in Non-standard Domains at the end of page $39$, they display, without explanations, that $\forall f \in H^k$ is homogeneous of degree $k$, $f=r^k \hat{f}$, where $\...
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$A^{\frac 32}$ for $A\geq0$ self-adjoint as an integral of the Resolvent

Let $A\geq0$ be a bounded self-adjoint operator on a Hilbert space. I would like to show that $$A^{\frac 32} =c \int_0^\infty A^2 (y+A)^{-1}y^{-\frac 12}\text{d}y,$$ where $c>0$ is an appropriate ...
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Homogeneous polynomials - Explanation of $f(x,y,z) \in P^k$, then $f(x,y,z)= \sum_{i \geq 0} f_i(x,y) z^{k-i}$ [closed]

In the document Spectral Geometry in Non-standard Domains at the end of page $37$, they display, without explanations, that if $f(x,y,z) \in P^k$, then $f(x,y,z)= \sum_{i \geq 0} f_i(x,y) z^{k-i}$. I ...
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1answer
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Find spectrum of integral operator

Let Af(x) = $\int_0^1 K(x,y)f(y)dy$, $A:L_2[0,1]\rightarrow L_2[0,1].$ Where $K(x,y) = \sinh(\min(x,y)\sinh(1-\max(x,y)). $ where $\sinh(x) = \frac{e^x - e^{-x}}{2}$ Find $\sigma(A), ||A||.$ I ...
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35 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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142 views

Weyl's asympotic law for eigenvalue : $\lim_{\lambda \to \infty} \frac{M(\lambda)}{\lambda}=\sum_p \frac{A(D_p)}{4 \pi}$

In the book Strauss W.A. Partial differential equations - An introduction (Wiley, $2008$, $1$nd Ed.) page $310 - 311$, it is probably a silly question, but is there anyone could give me a hint how to ...
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1answer
62 views

Find the spectrum of an operator

I am trying to learn some basic stuff about spectral theory, and I am a little bit lost. Please, could you help me and tell me how to find $\sigma(T)$ and $\sigma_p(T)$ of the operator $T:C([0,1]) \...
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29 views

Eigenvalues of block graph

Let us consider a graph $G$ having $m$ number of complete sub-graphs $K_{n_1},K_{n_2},...,K_{n_m}$ which have size $n_1,n_2,...,n_m$ respectively. Further $\forall i$, one vertex of $K_{n_i}$ is ...
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1answer
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Courant-Hilbert's Book: Weyl's asymptotic law for eigenvalues - Planar domains

In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment Now an arbitrary plane domain $D$ can be approximated by unions of ...
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Explanations : $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1)$ $∩$ $($first quadrant$)$

I am stuck on a problem for a good while now. Is there anyone could tell me rigorously why $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1) ∩ ($first quadrant$)$ of the problem of rectangle on page $18-19$ of ...
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If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give ...
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geometric interpretation of eigenfunctions of a vector field

Let $M$ be a smooth manifold and $X\in\mathfrak X(M)$ be a section on the tangent bundle. What is the geometrical interpretation of the eigenfunctions of $X$. That is functions in $f\in \mathcal C^\...
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The bigger the domain, the smaller the first eigenvalue - $\lambda(M_2) \leq \lambda(M_1)$ on the Laplacian

I know it is probably a silly question, but is there anyone could help me to complete of the corollary $3.1$ of that document? I pass a lot of time to try understanding the problem, but I can't ...
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1answer
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How could we obtain $\lim_{n \to \infty} \frac{\lambda_n}{n}=\frac{4 \pi}{ab}$?

Related to the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$, is there anyone could explain to me how is it ...
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1answer
24 views

The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
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Weyl's asymptotic law for eigenvalue on the rectangle $D = \{0 < x < a, 0 < y < b \}$ - $N(\lambda) \geq \frac{\lambda ab}{4 \pi} - C \sqrt{\lambda}$

I have a few difficulties understanding the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$. I've managed to ...
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35 views

Square summable solutions of second order difference equations

There should be a standard argument for the following claim: If the system of equations of the form $$a_{n-1}u_{n-1}+(b_{n}-z)u_{n}+a_{n}u_{n+1}=0, \quad n\in\mathbb{Z},$$ where $a_{n}\in\mathbb{C}\...
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1answer
29 views

real spectrum clarification

Let $T: X\to X$ be a bounded operator on the real Banach space $X$.Does the spectrum of $T$ consist of the reals in the spectrum of its complexification?Or are they the same thing by definition?
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1answer
36 views

Weyl's asymptotic law for eigenvalues - Rectangle $D = \{0 < x < a, 0 < y < b \}$

Let the domain $D = \{0 < x < a, 0 < y < b \}$ in the plane. We now that $$\lambda_{n,m} = \frac{n^2 \pi^2}{a^2}+\frac{m^2 \pi^2}{b^2}$$ with the eigenfunction $$u_{n,m}= \sin(\frac{nπ}{a}...
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33 views

Dirichlet conditions - Proof of theorem $4$ on an example

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
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1answer
28 views

If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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42 views

Dirichlet conditions - Explanation of the proof of theorem $4$

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...