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

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Spectrum of the circle - Clarification of certain points

I have a bit of difficulty as doing the problem Eigenvalues of the circle over the Laplacian operator. Since $g$ is a periodic function, do I have to use the Fourier series? If so, how could I do that?...
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134 views

Eigenvalues of the circle over the Laplacian operator

I would like to find the spectrum of a circle. It seems that there are no boundary conditions, but I'm not quite certain. How could I find the spectrum of a circle over the Laplacian operator?
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Wave trace on $1$-dimensional circle - How about the spectrum of this circle?

I have to find the wave trace for the Laplacian on the $1$-dimensional circle. Generally, the wave trace is defined (see this website) as $$W(t)= \int_{M} K_t(x,y)dy=\sum_j \cos(t \lambda_j)=\Re \...
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How to prove that $ e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...
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1answer
26 views

range of weighted shift operator

Consider the weighted shift operator on $\ell^2$ space defined by $T(x_0, x_1, x_2, ...) = (0, x_0, 2x_1, 3x_2, 4x_3, ...)$ with domain $$\mathcal{D}(T) = \{(x_n) \in \ell^2 : \sum_{n=0}^{\infty}|(n+...
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1answer
25 views

Element with maximum magnitude in $A \leq \max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A

Let $A$ be a matrix with real values. Is it true that element with maximum magnitude in $A$ is less than $\max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A? That is, is $$ \max_{ij} |A_{ij}...
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proof of an equality norm

Let the mapping $T:\ell^{2}\rightarrow \ell^{2}$ is defined as follow. $$T(x_1,x_2,\ldots,x_n,\ldots)=(x_1,\dfrac{1}{2}x_2,\ldots,\dfrac{1}{n}x_n,\ldots)$$ In this case, i've easily earned: $$\sigma(T)...
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40 views

Green's function on the unit interval $[0,1]$

I would like to find the Green's function on the unit interval $[0,1]$. We have to solve the linear equation $$\frac{d^2f(x)}{dx}=g(x)$$ with boundary condition $f(0)=a$, $f(1)=b$. So the Green's ...
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15 views

What does 'mode' mean in this context?

The spectrum of an operator $\mathcal{L}$ is the disjoint union of two sets: the point spectrum that consists of all isolated eigenvalues with finite multiplicity and its complement, which we call the ...
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31 views

Laplacian on the straight circle

I would like to well understand the notion of Green's function. I know that a Green's function, $G(x,s)$, of a linear differential operator $L = L(x)$ acting on distributions over a subset of the ...
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10 views

Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
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$U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$ - Extract the eigenvalues of the eigenfunction $U$?

On a certain problem, I obtain $U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$, where $J_s$ is the Bessel's function of order $s$, and we know by the Dirichlet boundary conditions that $U(r_0, \...
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44 views

Compute $\sum_{n=1}^\infty e^{\pi^2(\frac{n^2}{a^2})t}$ [duplicate]

I have to compute $$Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$$ with $\lambda_n=\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)$. So $$\sum_{m,n=1}^\infty \exp\left(-\pi^2\left(\frac{m^2}{a^2}+\frac{n^...
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39 views

Normal operators are unitarily equivalent iff they have the same spectrum

Here is a more general claim that implies the argument in the title: Show that if $S$ and $T$ are two normal operators, then there is a $*$-isomorphism $\pi: C^*(S)\to C^*(T)$ s.t. $\pi(S)=T$ iff $\...
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$g(y)=k_1 \cos (\sqrt{\alpha}y)+k_2 \sin (\sqrt{\alpha}y) $ - Value of $k_1$?

I have to find the constants $k_1$ and $k_2$ of the function $g(y)=k_1 \cos (\sqrt{\alpha}y)+k_2 \sin (\sqrt{\alpha}y) $ in using $g(0)=g(b)=0$. So I found that $k_1=0$. Furthermore, I think $\alpha = ...
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EDO by separation of variable - When are we certain that $\lambda \geq 0$?

If we use the separation of variables to solve $u_t-u_{xx}=-u$, $u=u(x,t)$, then we obtain $f(x)g'(t)-f''(x)g(t)=-f(x)g(t) \iff \frac{g'(t)}{g(t)}+1=\frac{f''(x)}{f(x)}$. The only way that a function ...
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1answer
25 views

Why is the Union of the Orthonormal Basis of $W$ and of $W^{\perp}$ an orthonormal Basis of $V$?

Let $V$ be a $\mathbb{K}$-vector space with a dot product. Let $T \in L(V)$ (endomorphism) be a self-adjoint operator ($\forall x, y \in V, T(x) \cdot y = x \cdot T(y)$) Let $W$ be an invariant ...
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relatively compact subspace

Can someone please help me to finf the answer for the following question: For $r\in \mathbb{N}$ ,we note $E_r=\{P\in \mathbb{R}[x_1,x_2,\ldots,x_d] \mid \deg P\le r \}$. and for $P\in E_r$ we ...
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19 views

Spectrum of a rectangle - Question

Question : Find the spectrum of a rectangle I know that the spectrum is the set of eigenvalues of the Laplacian within the region, with Dirichlet boundary conditions. For each $\lambda$, the ...
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21 views

Spectrum - rectangle, disk and the sphere

My supervisor asked me to find the spectrum of the rectangle, the disk and the sphere. Is anyone could give a bit of detail of what he means exactly? en.wikipedia.org/wiki/Spectral_shape_analysis. ...
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13 views

compacity of the resolvent

can someone please help me to solve the following problem: Let $A=-\Delta_V^{(0)}$ be an operator with domain $D(A)=C_0^{\infty}(\mathbb R^n)$(Witten Laplacian) which verifies : $\exists c\ge 0$ such ...
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20 views

Relation between the eigenvalues of $\Delta$ and counting lattice points

I was reading a paper with the following information: "Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the ...
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21 views

Essentially maximal accretive operator

Can someone please help me to prove the following equivalence: Let $A$ be an operator in $H$ (Hilbert space) with domain $D(A)$. $1)A$ is essentially maximally accretive. $2) A$ is maximally ...
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37 views

Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-...
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Witten Laplacian problem-show that $\lim\limits_{n \rightarrow +\infty} ||f_nu||=0$

Can someone help me please to correct my answer of this problem: We consider the Witten Laplacian with domain $C^∞_0(R^d)$. We know that this operator is essentially self adjoint,positive and with ...
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Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...
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41 views

Isospectral transformation of ODE

Is there a transformation of coefficients of differential operators w/periodic coefficients on the real line $a_n(x+2\pi)=a_n(x)$, that preserves the eigenvalues of their monodromy matrices? $$Df(x)=\...
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the “real spectrum” of an operator acting on a real Banach space

Let $X$ be a Banach space over the field $\mathbb{R}$, and denote by $\mathcal{L}(X)$ the space of continuous linear operators acting on $X$. The spectrum $\sigma(T)$ of an operator $T\in\mathcal{L}(...
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49 views

Approximate point spectrum of a normal operator

how can I show the following theorem? Let $H$ a Hilbert space and $T:H \to H$ a linear, continuos and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in \...
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Deriving spectral norm or similar quantity for structured random matrix

I have a problem where I have no idea to start. Suppose a simple Least Squares system with $M$ unknowns $c$ and $N$ observations $y$ which is given through the linear mapping $X$: $$y = X c$$ It is ...
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39 views

Finding an eigenvalue

I got the following statement to prove: Let $A \in \mathbb R^{n \times n}$ be a (column-)stochastic matrix, i.e. $A \geq 0$ (which means $a_{ij} \geq 0$ $\forall i, j \in \{1, \dots ,n\}$) and $\sum_{...
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Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition

Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one?
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42 views

Riemann mapping theorem - Personal project

I would like to work on the Riemann mapping theorem this summer. Does anyone could give me some good references linked to this objective. For your information, I now currently finishing a degree in ...
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49 views

Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for $M_\...
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1answer
20 views

separable weakly dense subalgebra of a von Neumann algebra

This is an exercise of "A short course on spectral theory" by William Arveson. For any von Neumann algebra $\mathcal{M}$ on a Hilbert space $H$, I need to show there exists a unital $C^\ast$-...
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1answer
21 views

Limit of sesquilinear forms is a sesquilinear form

Suppose $P_n$ is a monotone sequences of orthogonal projections in a complex Hilbert space $\mathcal{H}$. I want to show that the limit of the sesquilinear forms defined by: $\Gamma_n(x,y)=(x,P_n y)$ ...
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Monotone sequence of orthogonal projections on a complex Hilbert space

Suppose $P_n$ is a monotone sequence of orthogonal projections on a complex Hilbert space $\mathcal{H}$, i.e. $V_n= Im(P_n)$ is a decreasing or increasing sequence of subspaces and $P_n^\star=P_n$ and ...
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Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
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Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all $\...
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An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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51 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
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32 views

Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
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30 views

The Hilbert-Schmidt Theorem for Compact, Self-adjoint operators

Suppose $T$ is a compact, self-adjoint operator on a Hilbert space $\mathcal{H}$. Then there exists an orthonormal set $\{e_n\}_{n=1}^{\infty}$ of eigenvalues of $T$ such that every $x \in \mathcal{H}$...
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34 views

Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f'' $ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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22 views

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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1answer
29 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 ...