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

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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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60 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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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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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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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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97 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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71 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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369 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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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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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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162 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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290 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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111 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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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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339 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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157 views

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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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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226 views

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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108 views

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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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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28 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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31 views

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 ...
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Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
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Is there any relation between the Gershgorin circles of a matrix and its resolvent?

Let $A$ be a real symmetric matrix. Now fix a diagonal index say "i" and let $x > max-eigenvalue(A)$. Now is there any thing known about the Gershgorin circle of $[1/(x-A)]_{ii}$ in terms of the ...
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Spectra of operator matrices

Suppose we are given a bounded linear operator $A\colon X\to X$ on a Banach space which is injective and has closed range. Can we find two other operators $T$ and $S$ say such that ...
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The spectral projections of convolution operator

Given a self-adjoint operator $A$ in a Hilbert space $H$. How can one find its spectral projections $\{E_{\lambda}\}_{\lambda\in\sigma(A)}$? In particular, given a convolution operator on $L^2(G)$, ...
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Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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Intuition for Laplacian matrix of a graph's eigenvectors and eigenvalues

I am having difficulty finding intuition for Laplacian matrix eigenvalues/vectors in terms of non-regular, non-complete graphs. For example, consider the L, Laplacian, on a graph, G, a set of points ...
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50 views

Explicit inverse of $\lambda-U$ when $U$ is unitary and $|\lambda|<1$

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. By the spectral theorem, it is known that $\sigma(U)\subseteq \{z\in \mathbb{C}:|z|=1\}$. How can the explicit inverse of $\lambda-U$ be ...
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Reducing subspaces of a normal operator

If $A$ is a normal operator on an infinite dimensional Hilbert space $H$, then $H$ is the direct sum of a countably infinite collection of subspaces that reduce $A$, all with the same infinite ...
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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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Spectral radius of matrix from SOR method

Suppose we write a matrix $A = L + D + U$ with lower triangular, diagonal and upper triangular parts. When trying to solve the equation $Ax=b$, we use a successive overrelaxation technique such that ...
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25 views

Spectral convergence for collocation methods

Spectral methods work (simplified) as follows. Consider the problem \begin{align} \partial_t u(t,x) = \mathcal{L} u(t,x) \end{align} where $\mathcal{L}$ is some differential operator. We then try to ...
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Spectral Theorem for bounded, selfadjoint operators

I am trying to understand the proof of the spectral theorem for bounded, selfadjoint operators on a Hilbertspace $H$ in the book 'Functional Analysis' from Dirk Werner. The structure of the proof ...
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Can one go from eigenvectors (and eigenvalues) to the SVD of a matrix?

If I know all the eigenvectors and eigenvalues of a matrix, can I use that to directly write down the SVD of a matrix? (...of course one trouble is that for the $0$ eigenvalues of the matrix, the ...
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Relationship between spectrum of $-\Delta:H^1(M) \to H^{-1}(M)$ and $-\Delta:L^2(0,T;H^1(M)) \to L^2(0,T;H^{-1}(M))$?

Let us take a compact Riemannian manifold $M$. Let us define $-\Delta:H^1(M) \to H^{-1}(M)$ by $$\langle -\Delta u, v \rangle = \int_M \nabla u \nabla v$$ and $-\tilde \Delta:L^2(0,T;H^1(M)) \to ...
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For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
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Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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56 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
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49 views

Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
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63 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...