Spectral theory is a study of generalized notions of operators, eigenvalues and eigenvectors.

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Point spectrum of operator on $\ell^2$?

Considere the bounded linear operator $S:\ell^2\longrightarrow \ell^2$ given by $$ S(\xi_j)_j:=\left(\frac{\xi_2}{1}, \frac{\xi_3}{2}, \frac{\xi_4}{3}, \ldots\right).$$ How to show the point spectrum ...
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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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15 views

Spectrum of adjoint operators

If $A \subset \mathbb C$, we set $A^* = \{\bar z: z \in A\}$. I want to prove the following theorems. $\rho(T)^* = \rho(T^*)$ and $\sigma(T)^* = \sigma(T^*)$. $\sigma_c(T)^* = \sigma_c(T^*)$. ...
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64 views

Two definitions of spectrums

In Kreyszig's Introductory Functional Analysis Page 371, the point spectrum is defined as $\sigma_p(T)$ such that $R_\lambda(T) = (T - \lambda I)^{-1}$ does not exist. While in my functional ...
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34 views

resource on integral operators

Can you please suggest for me a good resource on integral operators.These are the specific topics that I am looking for: Bounded linear operators in hilbert space. Compact operators Spectral theory ...
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1answer
31 views

A continuous field of C* algebra, $C(\mathbb T)\rtimes\mathbb Z_2$

Given a $C^*$-algebra, $A=${$f:[0,1]\rightarrow M_2(\mathbb C)$ where $f(0),f(1) $ are diagonal } which is isomorphic to $C(\mathbb T)\rtimes\mathbb Z_2$, How can I determine its continuous field ...
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64 views

Eigen value and Regular Graph (not Strongly Regular graph).

$A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for ...
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35 views

A property of measurable functional calculus

A seemingly simple property of the measurable functional calculus: Let $A$ be a self-adjoint operator on a Hilbert space $H$ and let $P$ be the associated projection-valued measure, such that $A = ...
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1answer
34 views

Evaluate the spectrum of a bounded linear operator

$H$ is a separable Hilbert space over $\mathbb C$ and $\{u_n\}$ is a maximal orthonormal set of H. $A \in B(H)$ and there exists $\lambda \in \mathbb C$ such that $$A(u_n) = \lambda u_n - u_{n+1}, n = ...
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40 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
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12 views

Spectrum of multiplication operator and point masses

Consider the multiplication operator $A$ on $L^2(\mathbb R,d\mu)$, $Af = \lambda f$ for some function $\lambda \in L^2$. Then $f \in L^2$ is in $\ker(A-z)$ (for some $z \in \mathbb R$) implies that ...
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35 views

Spectral measure associated to eigenvector of self-adjoint operator

Let $A$ be a self-adjoint operator on .the Hilbertspace $H$ and let $\lambda_0$ be an eigenvalue of $A$ with corresponding eigenvector $\psi$. The spectral theorem tells us,that there is a ...
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16 views

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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1answer
41 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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21 views

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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41 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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33 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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2answers
74 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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14 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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1answer
20 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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21 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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36 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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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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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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14 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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1answer
25 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 ...
3
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1answer
28 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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24 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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42 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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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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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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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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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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1answer
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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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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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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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58 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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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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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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1answer
26 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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39 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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21 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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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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1answer
80 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
44 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
58 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 ...