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

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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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52 views

Spectrum of operator $Af(t) = \int_0^{t^2} f(s)ds$ on $L^2[0,1]$

Consider a linear operator $A\colon L^2[0,1]\rightarrow L^2[0,1]$ that acts as follows: $$Af(t) = \int_0^{t^2} f(s)ds$$ The problem is to compute its spectrum. I know that the operator is compact ...
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Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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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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46 views

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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154 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
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29 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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37 views

inverse of sum of diagonal matrix and eigendecomposition

I would like to simplify the following inverse computation : $$(D + A)^{-1}$$ where $A=U\Sigma U^T$ (eigenvalue decomposition). And D is a diagonal matrix I know the inverse of A is ...
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39 views

Determining primitive ideal space of C* algebra

What is the general way of determining the space of primitive ideals of the C* algebra if there is any? Thanks.
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61 views

What will happen if we try to reconstruct signal using phase only or magnitude only?

I am studying Fourier Transform and it's inverse. We get phase and magnitude from Fourier transform and reconstruct it back from both together My question is that What will happen if we try ...
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54 views

Convergence of spectrum with multiplicity under norm convergence

This article by Joachim Weidmann claims that, if a sequence $A_n$ of bounded operators in a Hilbert space converges in norm topology, i.e., $\|A_n - A\| \rightarrow 0$, then "isolated eigenvalues ...
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23 views

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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34 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
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18 views

Spectrum of Schrödinger operator in dimension two with periodic decreasing potential

I ask about what are the conditions on the potential $V$, if we have a discrete spectrum in $]-\infty,0[$ of $H(h)=H_{0}+V(y,hy)=-\Delta_{y}+V(y,hy)$, where $\hspace{0.25em}h\searrow 0$ $V$ is ...
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30 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 ...
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29 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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25 views

Approximate eigenvectors of the closure

Let $A_0$ be a closable operator on a Hilbert space and let $A = \bar A_0$ (i.e. $A = A_0^{\ast \ast}$). Let further $(f_n) \subset D(A)$ (domain of $A$) be an approximate eigenvector for $z \in ...
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44 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), ...
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47 views

time evolution of a density function

I have a time series of density functions, say A1-A5. Each density function is defined as $f(x)=\Sigma_{i=1}^{N} \beta(x-a_i)$, where $\beta$ is a smoothing function (e.g., gaussian or delta), and N ...
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7 views

Tighter bound than submiltiplicativity of spectral norms of matrices

Suppose A, B C are arbitrary matrices of appropriate dimensions, so that we can define $X = ABC$. Let $\| . \|$ be the spectral norms. Using submultiplicativity of the spectral norms, we can write ...
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64 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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31 views

What does it mean to take the spectral radius of a derivative operator?

I'm looking on the wiki page for a finite volume scheme that solves the following first-order equation: $$ \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} = 0 $$ The scheme just ...
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37 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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27 views

Mapping properties of differential operators: Reference for targeted reading

In my studies (currently I am trying to understand spectral properties of differential operators) I am encountering operators that are unbounded. To be more concrete, here is an example that I ...
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94 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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96 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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51 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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22 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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58 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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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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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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45 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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24 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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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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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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62 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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104 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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1answer
54 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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1answer
95 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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51 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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70 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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Equivalent definition of the Kreiss constant

Let $A$ be an $n\times n$ matrix. The $\epsilon$-pseudospectrum of $A$ is defined to be $\{z\in\mathbb{C}:\|(zI-A)^{-1}\|\geq\frac{1}{\epsilon}\}$, where the norm considered is the operator norm ...
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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 ...
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1answer
36 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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136 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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2answers
70 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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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 ...
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29 views

Self adjoint operator with countable eigenvalues

It has been explained here that any self adjoint operator on a seperable Hilbert space inhabits at most countable many eigenvalues. Now I wonder, can one construct an operator $$ T : L^2([0,1]) \to ...
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11 views

Relationship between pseudospectral abscissa and inverse of the norm of the resolvent

For $\epsilon>0$, the $\epsilon$-pseudospectrum $\Lambda_{\epsilon}(a)$ of an element $a$ of a Banach algebra $A$ is defined to be ...
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30 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 ...