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

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Calabi's theorem

I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. ...
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Proposed proof Decomposition theorem

Let $\mathcal{A}$ be a unital Banach algebra. I want to prove that if $a \in \mathcal{A}$ and spectrum $\sigma(a) = \sigma_{1} \cup \sigma_{2}$ where $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
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34 views

Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
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Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
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How to extract power of top two frequencies of a spectrum without using an FFT?

What I'm trying to do is see if a particular frequency component becomes dominant (and I don't really know what the dominant frequency is). Therefore, I figured that I can get the top two peaks of ...
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76 views

Spectral theorem for compact normal operators

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting ...
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51 views

Deficiency indices for differential operator on half-line

1) What is the domain of the adjoint $A^\ast$ of the differential operator $Af = i \frac{d}{dx}$ with $D(A) = \mathcal C^\infty_c (0,\infty)$? 2) I want to compute the deficiency indices of $A$. By ...
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120 views

Clustering with SVD

I'm trying to do some clustering on a graph, which is represented by an adjacency matrix $B = A^2$, where $A$ is symmetric. I tried several methods like taking the eigenvectors of the Laplacian $L = ...
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46 views

A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
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41 views

About prime geodesic cycles and deck transformations group

I'm proving theorem 2 occurring in Sunada's paper Riemannian coverings and isospectral manifolds. Unfortunately Sunada's quotes himself to the following paper: Tchbotarev’s density theorem for closed ...
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77 views

Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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69 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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59 views

Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
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Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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88 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
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39 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
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112 views

interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
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80 views

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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524 views

Inverse of bounded self adjoint operator on HS is self adjoint?

Let $A=A^{*}$ be a bounded self adjoint operator on a Hilbert space $\mathcal{H}$ with Range Ran$(A) = D$ dense in $\mathcal{H}$. $A$ is injective, since Ran$(B) \perp ker(B^{*}) = ker(B)$. So ...
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91 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
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79 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
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181 views

Application of the spectral decomposition theorem to PDE

In the compact version, there is many application of the spectral decomposition of a bounded self-adjoint operator: Sturm-Liouville, spectre of the laplacian,... But for the general version which ...
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310 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
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Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
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75 views

Calculus of Variations statement of a Singular Value Decomposition?

My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
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117 views

Dirichlet eigenvalue problem on the Hilbert cube

I'm trying to solve the Dirichlet problem for the Helmholtz equation \begin{aligned}-\triangle u & = & \lambda u, & x\in\Omega,\\ u & = & 0, & x\in\partial\Omega, ...
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72 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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385 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
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Diagonalization discrete laplace operator

How do i diagonalize the discrete laplace-operator $\triangle$ on $l^2(\mathbb{Z})$ (defined by e.g. $\triangle e_k = e_{k-1} + e_{k+1}$, with $e_{k}$ being the canoical basisvectors of ...
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$L^p$ norm of diagonal is $\leq$ Schatten $L^p$ norm

Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove ...
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119 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
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Examples for spectrum of an operator

Looking for easy-to-understand examples for the spectrum of an operator, preferably so that they exposed some special properties. The right shift is a nice example of an operator which does not have a ...
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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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Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
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Can one hear the *material* of a drumhead?

"Can one hear the shape of a drum?" is a well known problem, originating from Kac, 1966, that questions whether an (idealized) drum head is completely specified by its spectrum. That is: is the ...
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References on the spectral theory of the Floquet problem

The Floquet problem is about the linear ordinary differential equation $$ \dot{\psi} = A(t) \psi. $$ Here $A(t)= A(t+T)$ is a periodic $n\times n $ matrix. Suppose $A(t) =-i H(t)$ with $H(t)$ being ...
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Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...
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Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on ...
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Spectrum radius upper bound of Hadamard product

I am doing some research that relies on Hadamard product of two matrices bound, the most famous one that I encounter is : $\rho(A\circ B)=\rho(A)\rho(B)$ this seems to be trivial when I test it with ...
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Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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Spectral Representation of a Linear Combination of Sinoids

Let $\left\{X_t\right\}$ be the process defined by $$ X_t=\sum_{j=1}^nA(\lambda_j)e^{it\lambda_j} $$ in which $-\pi<\lambda_1<\lambda_2<\ldots <\lambda_n=\pi$ and ...
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Finding the spectral representation of the Delta function given the Green function of an operator

I'm working through the problems of a book on Sturm-Liouville problems. In a problem I found the Green function for the following SLP2 problem $$\frac{-d^2g}{dx^2}-\lambda g=\delta(x-\xi)$$ ...
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Eigenvalues of a certain product of matrices with special structure

Sorry for cross-posting from MO. Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the ...
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Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
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A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
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Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
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Subspace Perturbation

For two positive semidefinite matrices $A,B\in\mathbb{R}^{n\times n}$, with dominant $r$ dimensional subspaces $U,V\in\mathbb{R}^{n\times r}$ and eigenvalues $\Sigma_A, \Sigma_B$, what can we say ...
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Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...