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

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Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
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78 views

Eigenvalues of the operator $(Tu)(x)=\int_0^x (\int_t^1 u(s)ds)dt.$

Consider the linear operator $T$ in $L^2(0,1)$ defined by: $$(Tu)(x)=\int_0^x \left(\int_t^1 u(s)ds\right)dt.$$ I have managed to prove that it's continuous,self adjoint,compact but now I have to ...
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39 views

Properties of the operator $ (Tu)(n)=u(n+1)-u(n-1)$

Let $T:l^2(Z,R)\to l^2(Z,R)$ the linear bounded operator defined by: $$ (Tu)(n)=u(n+1)-u(n-1)$$ a)Prove that the image of T is dense. b)Prove that $\forall \lambda\neq 0\quad t-\lambda I$ is ...
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70 views

Connections between Lebesgue-Radon-Nikodym decomposition and spectral decomposition

Perhaps this is a silly question to ask but it's been on my mind for a bit. When I took my first course in functional analysis a year ago, we covered spectral theory. Particularly, we covered spectral ...
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33 views

Does non-Hermitian implies at least one complex eigenvalue?

Ok so I'm studying linear algebra and we went trough the Spectral Theorem, including and proving the fact that for every Herimitian matrix, its eigenvalues have $0$ imaginary parts. I was wondering is ...
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286 views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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100 views

Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
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85 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
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67 views

Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
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486 views

The eigenvectors of a matrix and its transpose that correspond to the same eigenvalue are not orthogonal

Spent hours trying to prove this after encountering it in Lax's discussion of the spectral theorem, but no luck. Here's the problem (it is Theorem 18 in Lax 2ed, Chapter 6): A mapping $A$ has ...
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62 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
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138 views

Spectrum of a weakly compact operator

It is well known that the power of a weakly compact operator is compact. Is the spectrum of a weakly compact operator is the same as a compact operator?
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335 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
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33 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
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107 views

expectation of norm of orthogonal projector

The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace $T := span\{U1, U2\}$ where $U1$ is a random vector uniformly distributed over unit ...
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57 views

Spectral theorem for $n$-tuples of selfadjoint operators

I need a 'good' reference to the following version of the Spectral Theorem: Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on ...
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86 views

change of spectrum under diagonalisation

I have the following question. Let $T\colon \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on a Hilbert-space $\mathcal{H}$. By spectral theorem we know that there exists a measure ...
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132 views

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ...
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52 views

Proof of a Lemma guaranteeing the existence of the Borel-measurable functional calculus

In my lecture I had the following Lemma, which guarantees the existence of the Borel-measurable functional calculus: Le $(H,<,>)$ be a complex Hilbert space and let $q:H\rightarrow \mathbb{C}$ ...
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58 views

Using Weyl sequences to prove relation between quadratic form and spectral radius

I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$ holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula ...
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25 views

Time Series: Spectral Density

Prove that $$ \int_{-\pi}^\pi e^{i(k-h)λ} dλ = 0 $$ if k!=h solving the integral gives $$ \frac{2 sinh(\pi * i(k-h))}{i(k-h)} $$ I don't see how this will give 0 with k!=h (also if k=h, the ...
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122 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
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1answer
128 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
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37 views

Do eigenfunctions from symmetric ODEs really make up a base for $L^2$?

I'm confused about a theorem in some hand-out material on spectral theory for ODEs. The BVP is stated in the form $$ \begin{cases} L u = f\\ R u = 0 \end{cases} $$ over $I = [a,b]$. And ...
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66 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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61 views

Asymptotic of the heat kernel

I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian on a Riemannian Manifold" and not quite clear how to get the estimate $(4\pi t)^{n/2}|Q_k * H_k|\leq C \cdot t^{k+1}$ on a compact ...
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225 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
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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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For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
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Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
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62 views

Spectral radius of nonnegative matrices with an M-matrix difference

Let $A$ and $B$ be non-negative matrices (both with real spectra) and $A−B$ is an M-matrix. Are these conditions necessary to prove that the spectral radius of $A$ is greater than the spectral radius ...
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Solving an inverse spectral problem

In order to solve the inverse spectral problem: $$ -y''(x)+q(x)y(x)= \lambda _{n}y(x) $$ If we want to obtain $ q(x) $ what we should need about the spectrum? a) The eigenvalue staircase $ ...
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Examples of deeper results in finite-dimensional vector spaces?

this one is a bit inverted! So I am busy doing an advanced undergrad course in Linear algebra, and it is going very well, the problems in the book seem fairly routine. To be able to see if I am any ...
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How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
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191 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
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113 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
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158 views

How does determinant relate to argument of matrix function?

I have a matrix function $R \mapsto J(R)$ from $\mathbb{R}$ to the set of irreducible matrices with non-negative entries. We can assume that $J(R)$ is $d \times d$, although any solutions that work ...
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The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$

On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
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Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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243 views

“Algebraic multiplicity” for eigenvalues of a Sturm-Liouville-like problem?

Following Coddington-Levinson's book Theory of ordinary differential equations, chapter 7: "Self-adjoint problems on finite intervals", let us consider the eigenvalue problem $$\pi(l):\begin{cases} ...
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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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138 views

Show that an operator is bounded (from Reed and Simon)

I am currently reading Reed and Simon's IV: Analysis of Operators, Volume 4 (Methods of Modern Mathematical Physics). I don't understand something they do in Theorem XIII.64. The problem is: Let $A$ ...
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376 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
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46 views

The trace of an integral equation?

I am reading a paper about spectroanalysis and encountered the following integral equation: $$\int_{-1}^{1}\frac{\sin A(x-x')}{\pi(x-x')}\psi(x')dx'=\lambda\psi(x)$$ Then the paper gives without proof ...
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75 views

Spectrum of an element

I'm having a little trouble calculating the spectrum of an element: specifically, the element $f(x) = \frac{1}{x}$, as an element of the bounded continuous functions from $[1, \infty)$ with pointwise ...
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38 views

Spectrum and tower decomposition

I'm trying to read "Partitions of Lebesgue space in trajectories defined by ergodic automorphisms" by Belinskaya (1968). In the beginning of the proof of theorem 2.7, the author considers an ergodic ...
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201 views

Study of the Laplacian on the Hyperbolic plane

What's a good reference for the simplest case? I'm interested in the spectral theory of the Laplace-Beltrami operator on the upper half plane (domain, self-adjoint extension, etc.). I only need this ...
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128 views

Prove a bilinear operator is symmetric and positive definite

I'm having problem showing the following: All operators are defined on $V$ which is real (not complex). Let $f$ be a bilinear operator that is anti-symetric (meaning $f(a,b)=-f(b,a))$ and let $J$ be ...
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2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...