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

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Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
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121 views

Closure of numerical range contains spectrum

Let $A: D(A) \subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator on a Hilbert space $\mathcal{H}$ with adjoint operator $A^{*}$. Given that $D(A) = D(A^{*})$ I'm trying to show that the ...
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136 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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160 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
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285 views

Spectral family property: $\lambda \geq M \Rightarrow E_{\lambda} = I$

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and I'm trying to follow his proof about some properties of a spectral family associated with a bounded self-adjoint ...
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109 views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
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334 views

Residual spectrum is empty

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468) For a bounded self-adjoint linear operator ...
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69 views

Proof that $A^n = \frac{1}{2\pi i} \oint _C z^n \left(z - A \right)^{-1} dz$

I'm working my way through Martin Schechter's 'Principles of Functional Analysis' (2nd ed.) and am trying to understand his proof of the following theorem, given on page 136: "Let $T:X\to X$ be any ...
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138 views

Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
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255 views

Compact resolvent

Given that the operator $$ Hf(x) = -xf''(x) + (x - 1)f'(x) $$ on the Hilbert space $L^2([0,\infty),e^{-x}dx)$ possesses, for each $n \in \mathbb{N}$, an eigenvalue $\lambda_n = n$ with eigenvector ...
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204 views

What is the spectral radius of the operator $T_k:C[0,1]\to C[0,1]$ defined as $T_k x (t)= \int_0^1 k(t,s) x(s) ds$?

I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous. And ...
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265 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
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42 views

Tensoring Spectral triples that are composed from Real algebras.

I have a misunderstanding that I am hoping is really quite trivial. In connes standard Non-commutative geometry model of electroweak interactions he takes the algebra input in his finite spectral ...
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242 views

Fixed-point method in many-dimensions

A well known method of easily solving multi-dimensional non-linear root finding problems, is to bring the equations into the form: $$\bf x = g(x)$$ And then iterating. The problem is, one has to find ...
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can I decide the following from the inverse of I-A?

I have a square matrix $A$. Is it possible to determine if its largest eigenvalue is smaller (by magnitude) than 1 by inspecting the matrix $(I-A)^{-1}$? (we can assume that $I-A$ is invertible.) ...
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197 views

Characterizations of the form domain for unbounded selfadjoint operators

This question follows from this one and especially from Willie Wong's answer: link. In Reed & Simon's book Methods of modern mathematical physics, vol. I, pag.277, the form domain of a ...
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357 views

Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is ...
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50 views

Isolated Eigenvalues on the Extensions.

I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ...
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21 views

Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$

Let $V$ be a complex inner product space. Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$? If not, ...
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77 views

If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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25 views

How to do spectral decomposition?

I missed the last couple classes due to a family emergency and am trying to catch up with review questions. However, I can't seem to find an online source that teaches how to compute a spectral ...
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35 views

Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ...
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42 views

Laplace-Beltrami operator

I'm interesting in the Laplace-Beltrami operator on a sphere, more precisely its spectral properties including the spectral function, etc. So if someone can give me some references that treats this ...
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31 views

What theorem is this in the infinite dimensional case?

The spectral theorem for compact, self-adjoint operators is as I have understood the infinite dimensional case for orthogonal diagonalisation of a symmetric case in linear algebra? But in linear ...
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1answer
47 views

Estimate spectral radius of operator product

In my research problem, I have to estimate the spectral radius of the following operator $\chi A$ where $\chi$ is a scalar function taking values 0 or 1 and $A$ is an operator. I can compute ...
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1answer
39 views

The distribution solution to $L^{+} u=0$ is u=0, where $L^{+}=-\frac{d}{dx}+x$?

Consider the creation operator $L^{+}=-\frac{d}{dx}+x$. If $u\in L^2(\mathbb{R})$ and is a distribution solution to the equation $L^{+}u=0$, then for any $\phi\in C_0^{\infty}(\mathbb{R})$ we have ...
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Calabi's theorem

I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. ...
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operator ideals and their relationship to the geometry of Banach spaces, and other questions [closed]

Albrecht Pietsch wrote an excellent book on operator ideals in 1978, which I use frequently. But, I was reading the preface to his book, and I do not understand it. He writes (in English ...
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Proposed proof Decomposition theorem

Let $\mathcal{A}$ be a unital Banach algebra. I want to prove that if $a \in \mathcal{A}$ and spectrum $\sigma(a) = \sigma_{1} \cup \sigma_{2}$ where $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
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1answer
45 views

Spectral theorem for momentum operator

I'm trying to apply the spectral theorem to the explicit example of the momentum operator: \begin{equation} p:= i\frac{d}{dx} \end{equation} on the domain $D=\{\psi\in H^1(0,2\pi)\ |\ ...
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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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Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
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Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of ...
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How to extract power of top two frequencies of a spectrum without using an FFT?

What I'm trying to do is see if a particular frequency component becomes dominant (and I don't really know what the dominant frequency is). Therefore, I figured that I can get the top two peaks of ...
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51 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
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Spectral theorem for compact normal operators

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting ...
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1answer
63 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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51 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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120 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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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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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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53 views

Application of the spectral mapping theorem

Let $T:L^2((0,2)\rightarrow L^2((0,2))$, $(Tx)(t):=\begin{cases} x(t+1), & 0<t<1\\ 0,& \text{elsewhere} \end{cases} $ Show that $T$ is well defined and $\sigma(T)=\sigma_p(T)=\{0\}$ ...
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Unitary Equivalent of Derivative in Fourier Space

It is known that for $L^2(\mathbb R)$ the operator $Tf(x) = if'(x)$ is unitary equivalent to $\hat T \hat f(\xi )= \xi \hat f(\xi) $. Where domain of T is $H^1(\mathbb R)$. Hence the Spectrum of T in ...
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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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128 views

Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
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68 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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84 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
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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 ...