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

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Are the solutions of a Sturm-Liouville equation entire in the spectral parameter?

In $[1]$ the following (paraphrased) claim is made: Let $q\in L^1_{loc}([0,\infty);\mathbb{R})$, and suppose $\varphi$ and $\theta$ solve the one-dimensional Schrödinger equation \begin{equation} ...
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38 views

Spectral projection and isolated point of spectrum

Let $u\in B(H)$ be a normal element with spectral resolution of the identity $E$ and $\lambda$ be an isolated point of spectrum $u$. Show that $E(\lambda)H = \ker(u-\lambda)$ . I can show that $E(\...
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36 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
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65 views

A paradox derived from the open mapping theorem

The problem comes from Erwin Kreyszig's Introductory Functional Analysis with Applications, section 7.4, problem 4: Let $T:l^2\mapsto l^2$ be defined by $y=Tx, x=(\xi_j), y=(\eta_j), \eta_j=\...
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101 views

How is the Point Spectrum of a Compact Operator Countable?

I'm working on understanding a proof that if an operator $A$ on a Hilbert space $\mathcal{H}$ is compact, then show that $\sigma(A) - \{0\} \subseteq \sigma_p(A)$. If you're not familiar with this ...
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1answer
42 views

spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...
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141 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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193 views

Find the spectrum of the operator $T: \ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined by $(Tx)_n = \frac{x_n}{n}$

Consider the linear operator $T:\ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined as $$ (Tx)_n = \frac{x_n}{n}, \quad x \in \ell^2(\mathbb{C}). $$ I can show that it is bounded with norm $\|T\|=1$, ...
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52 views

Making it possible to do a Fourier transform on it: $\frac{1}{(k+w)^2(a^2 +w^2)}$

Sorry for all the edits, I'm very stressed and not so used to Latex. Full question: consider a filter with impulse response $$h(t)=e^{-bt} u(t)$$ where $u$ is the unit step function. The input ...
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81 views

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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118 views

Efficiently compute the eigenvectors of the Laplacian of a symmetric positive matrix

I am working with a matrix A relatively large (200k x 200k), and I want to compute the eigenvectors of the Laplacian: $L = D - A^2$, where $A$ is symmetric. I don't need all eigenvectors, just a few ...
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59 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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42 views

adjoint of an operator. on $L^2(0,1)$, $Bf(x)=\int_0^x f(t)dt$

I see that the above operator is bounded. I ended up with an argument to calculate the adjoint as follows, $$ <f,Bg>=\int_0^1\overline{f(x)} \int_0^xg(t)\,dt\,dx $$ I see $f(x)$ as the ...
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52 views

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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31 views

Extension of a self adjoint Operator

Suppose we have a open (bounded) domain $\Omega$ in $\mathbb R^d$. And let a plane $\mathcal P$ in $\mathbb R^d$ divides the domain in two (disjoint) open sets. (say $\Omega_1$ and $\Omega_2$) Hence $...
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48 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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42 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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89 views

Discrete and Essential spectrum of Laplacian in $\mathbb R_{+}$ (with weird boundary conditions)

I am given on Hilbert Space $\mathcal H=L^2(\mathbb R_{+})$ $$ Af(x)=-f''(x) $$ and Domain of A is $$ D(A)=\{f\in H_2(\mathbb R_{+})\;\;| \;\;f'(0)+\alpha f(0)=0\} $$ for some $\alpha \in \...
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221 views

Convergence of the spectrum under norm resolvent convergence

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 norm resolvent sense. Since $A_n \to A$ in strong ...
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79 views

Relation between residual spectrum and point spectrum.

Suppose T is a bounded operator on a Hilbert space. Show that if λ is in the residual spectrum of T, then $\bar{λ}$ is in the point spectrum of the adjoint. Here is what I think needs to be done. ...
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322 views

When the point spectrum is discrete?

Are there some criteria to tell when the point spectrum of a linear operator is discrete? In general it is not the same (take the spectrum of the "annihilation" operator). More specifically, what are ...
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95 views

bounds of spectral radius of hadamard product

In Topics in matrix analysis by R. A. Horn, C. J. Johnson it's shown that for two positive matrices $A$ and $B$, $\rho(A\circ B)<\rho(A)\rho(B)$. I'm wondering whether the extension of this to ...
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1answer
150 views

Eigenvalue of the sum of a symmetric matrix and the outer product of it's eigenvector

I have a symmetric matrix $A$ with eigenpairs $(\lambda_k, v_k)$ with $k \in (1,..,n)$. A new matrix $B$ is made from an eigenpair $(\lambda_i, v_i)$ like this: $$B = A - \lambda_i v_i v_i^T$$ where $\...
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173 views

Several questions about integral operators.

I have been fumbling with expressions of the form \begin{equation} A\{f\}(s) = \int A(s,t)f(t)\operatorname{dt} \tag{$\star$} \end{equation} as a generalization of the matrix product. When looking ...
2
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60 views

Heat operator formalism via spectral projections and Dirac measure

I am currently reading very helpful notes on the Heat kernel $p(t,x,y)$ on a Riemannian manifold $M$ -- there is one aspect though that I am not sure I understand notationally, it says that we can ...
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149 views

Why the equality of spectral zeta functions imply the isospectrality?

Let $\Delta_{M_1}$ and $\Delta_{M_2}$ be the Laplace-Beltrami operators on two compact and connected Riemannian manifolds $M_1$ and $M_2$ respectively. We define the spectral zeta function (or ...
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104 views

How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Let $m$ be a positive integer and define $$ Lf = -\frac{d}{dx}(1-x^{2})\frac{df}{dx}+\frac{m^{2}}{1-x^{2}}f $$ on the domain $\mathcal{D}(L)\subset L^{2}(-1,1)$ consisting of all twice ...
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48 views

Question on the spectral radius, regular splitting, and non-singularity/non-negativity

Given $A$ in $R^{nxn}$ and its regular splitting M and N (A = M - N), $M$ is nonsingular and $M^{-1}$ and $N$ are nonnegative. If the spectral radius $p(M^{-1}N)<1$, show $A$ nonsingular and $A^{-...
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39 views

Notion of eigenspace preserving transformations

I was reading through a linear algebra textbook, and I'm not sure what does the following statement refer to: $\textbf{"For a vector space V, the eigenspaces of a transformation}$ $T: V \rightarrow ...
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1answer
66 views

Isolated eigenvalues of a self adjoint operator

If $X$ is a separable Hilbert Space and $T : X \to X$ selfadjoint and bounded, then the point spectrum $$ \sigma_p(T) $$ is only countable as explained here. I have the following three questions: ...
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71 views

GNS Construction on non-unital algebra

STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class $ξ$ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If $A$ is ...
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Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
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1answer
200 views

Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a function: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad f(\mathbb{R})\subseteq\...
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277 views

Spectral Measures: Stone's Formula

Hilbert Space: $\mathcal{H}$ Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Spectral Measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad H=\int\lambda\mathrm{d}E(\...
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Product unbounded operators

Let $A : D(A) \subset H \rightarrow H$ be unbounded and $B$ be a bounded operator, both of them are self-adjoint, then $(AB)^* = B^*A^*$ and $(BA)^* = A^*B^*$, right? I just wanted to be sure that ...
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1answer
279 views

Spectral theorem for unbounded self-adjoint operators, questions about the proof

I want to understand the proof of the Spectral theorem for unbounded self-adjoint operators. First the theorem: Let H be a separable complex Hilbert space, $A:D(A)\subseteq H\to H$ a densily defined ...
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Existence of idempotents versus existence of projections in a C*-algebra

Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections? Clearly, when $x$ is ...
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68 views

Is the minimiser of the quadratic form of a semi-bounded self-adjoint operator an eigenstate?

I am wondering whether the following fact, for which I know well the proof when $H$ is a Schroedinger operator (see Lieb-Loss, Analysis, Chapter 11), is also true in the general setting used below, ...
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54 views

Stieltjes inversion formula

Let $[a,b] \subset \rho(T)$ and $T$ be a self-adjoint operator then I want to show that $0=\frac{1}{\pi} \lim_{\varepsilon \downarrow 0} \lim_{\delta \downarrow 0} \int_{a+\delta}^{b+\delta} Im(\...
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1answer
113 views

Proof of the spectral theorem

I am currently going to through my proof of the spectral theorem that we had in class, but I feel that I have copied some nonsense from the board. So we defined the Cayley transform $U= (T-i)(T+i)^{-...
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Approximate Identities

Statement : Let $U$ be a C* algebra and$\lambda=\left\{A\in U:A\geq 0, ||A||<1\right\}$. If $B\in \lambda$ then if $X\in U$ $$||X^*(I-B)^2X||\leq ||X^*(I-B)X||$$ For reference this is from ...
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Discrete Laplace: ONB

Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ Regard the unitary transformation: $$U:\mathcal{L}^2(-...
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154 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
2
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1answer
210 views

Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- T)^{-1}:\quad\|R(\lambda)\|\geq\frac{1}{d(...
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Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
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40 views

What is the notion of “character” in the context of Cayley graphs?

I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to $\...
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40 views

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 L^2(0,...
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1answer
74 views

The second smallest eigenvalue of a complete binary tree

Apparently it is true that the second smallest eigenvalue of a complete binary tree is $\theta(\frac{1}{n})$. Can someone point out a reference which proves this?
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1answer
48 views

C* Algebra Positivity

STATEMENT: This is a proof from one of Qiaochu's notes on $C^*$ algebras. Proof: Let A be a $C^*$ algebra.We now want to show that for any $c\in A$ we have $c^*c\geq 0$. Suppose otherwise.We know ...