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

learn more… | top users | synonyms (1)

2
votes
0answers
60 views

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 ...
1
vote
2answers
35 views

Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the ...
0
votes
0answers
45 views

Symbol of an inverse on an invariant subspace of a certain elliptic operator

I have little experiences in dealing with pseudodifferential operators, so I hope I could get some hints for the following question: Let $L$ be an 2nd-order elliptic operator on $\mathbb{T}$ with ...
1
vote
0answers
22 views

Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
2
votes
1answer
116 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: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
3
votes
1answer
166 views

Properties of solutions to the associated Legendre ODE

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
2
votes
1answer
178 views

Spectrum of left shift operator: take two

This is my second attempt at calculating the spectrum of the left shift operator on a Hilbert space. I got stuck again and I would be grateful if someone could help. (You can find my previous (failed) ...
0
votes
0answers
41 views

Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
1
vote
1answer
11 views

Joint spectral radius of $\sigma( \mathcal A)$ and $\rho(A) < 1 \forall A \in \mathcal A$

Given $\mathcal A \subset R^{n \times n}$. The joint spectral radius is by: $$\sigma( \mathcal A) = \limsup_{m \rightarrow \infty}\sup_{A \in \mathcal A^m}\rho(A),$$ where $\rho$ is the normal ...
4
votes
0answers
46 views

What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
0
votes
1answer
26 views

Lower bound for the norm of the resolvent

I need to prove next statement (I want to do it for general case) $\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}$ I think it could be like this let $a\in \sigma(A) z ...
2
votes
1answer
76 views

Spectrum of left shift operator $L\in B(H)$

Let $H$ be a Hilbert space with an orthonormal base $e_i$ and $L$ the left shift operator $L\in B(H)$: $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$. I computed the spectrum could someone please tell ...
1
vote
0answers
31 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
3
votes
1answer
68 views

Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
1
vote
1answer
60 views

Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
3
votes
2answers
110 views

Difference between an eigenvalue and a spectral value

What is the difference in the definition of a spectral value and an eigenvalue. My notes from functional analysis says $\lambda$ is an eigenvalue of an operator $A$ if $\,\exists \, x \in ...
1
vote
2answers
44 views

Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
1
vote
0answers
53 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
1
vote
1answer
211 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
1
vote
2answers
130 views

Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
1
vote
0answers
48 views

Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
0
votes
1answer
38 views

Spectral theorem question

I am trying to understand how to develop the spectral measure of a bounded self-adjoint operator on a Hilbert space. For every continuous function on its spectrum, $f: C(\sigma(A)) \to \mathbb{C}$, ...
4
votes
1answer
86 views

Sturm-Liouville problem and periodic boundary conditions

I was wondering about this: I know that if a 1-d Sturm-Liouville operator is limit circle or limit point then the eigenvalues are simple ( so no degenerated spectrum). But in the case of periodic ...
0
votes
0answers
51 views

Relation between weighted outer product and unweighted outer product of a matrix

I know this may be quite straightforward, but I don't know how to prove it because it was quite a while since I took undergraduate linear algebra classes. Assume that an $n\times n$ real symmetric ...
4
votes
1answer
83 views

Computation the different spectrums of an operator

Let $X:=C^0([0,2],\mathbb C)$,$\phi\in X$ and $T\in L(X)$ defined as: $$(Tf)(t):=\phi(t)f(t),t\in [0,2]$$ Compute: $\sigma_p(T),\sigma_c(T),\sigma_r(T),\sigma(T)$ and $\rho(T)$ I am quite new to ...
2
votes
3answers
55 views

Can an operator have spectrum consisting of just one point?

Could it happen that an operator has a spectrum consisting of just one point? Sorry if this is really short. Was just pondering to myself random questions and this one came up. The good news is ...
0
votes
1answer
33 views

Are self-adjoint / Hermitian operators necessarily orthogonal / unitary?

I feel like self-adjoint / Hermitian operators are the "best" operators, since an operator that is self-adjoint can be orthogonally diagonalized, according to the Spectral Theorem (over the complex ...
0
votes
2answers
35 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
0
votes
1answer
39 views

Spectral Theory of an operator

If we define the spectrum of a bounded linear operator $T$ by $$\sigma(T)=\{\lambda\in \mathbb C:\ T-\lambda I \ \text{ has no inverse} \}.$$ What about $\sigma(T^{-1})$?
2
votes
2answers
157 views

Eigenvalue of a unilateral shift operator

Let $S:H\to H$ be a unilateral shift operator. I preferred in Example2.3.2 of Murphy's C*-algebras and operator theory that S has no eigenvalues. While $\{\lambda \in \Bbb C ; |\lambda|<1\} \subset ...
5
votes
0answers
28 views

Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
0
votes
1answer
22 views

Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
0
votes
1answer
24 views

Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
2
votes
1answer
33 views

As $\lambda \to \infty$, $||R_{\lambda}(T)|| \to 0$

"Let $T \in B(X,X)$. Prove that $||R_{\lambda}(T)|| \to 0$ as $\lambda \to \infty$." We have that $R_{\lambda}(T)x=(T-\lambda I)^{-1}(x)$. The problem doesn't specify but the book says that they will ...
1
vote
1answer
40 views

Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
0
votes
1answer
74 views

Spectral Measures: Spectral Spaces (II)

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: ...
0
votes
0answers
16 views

Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...
-1
votes
3answers
108 views

Spectrum of Multiplication Operator $T$ in $C[0,1]$

"Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$." This is an exercise from a PDF of notes I found ...
0
votes
1answer
75 views

Proof that the spectrum of an element of a Banach Algebra is non-empty

I don't see why the line indicated with ***** in the following proof is true in the proof that spectrum of an element of a Banach Algebra is non-empty (Arveson, p.27) : For every $\lambda_0 \not\in ...
0
votes
1answer
53 views

Spectrum of Unitary Operators

Let $T_1$ and $T_2$ be two unitary operators. Is it true that the spectrum of $T_1+T_2$ is contained in the closed disc of radius 2?
0
votes
0answers
35 views

Spectral methods with linear programming

Is it possible to model and solve some fundamental spectral methods (say Singular-Value Decomposition) with (Integer?) Linear Programming? Update: say you want to do SVD. Can you model it as a ...
1
vote
3answers
307 views

Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
0
votes
0answers
17 views

pertubation of an operator by orthogonal projection

Let $G$ be an operator with compact resolvent on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$ My question ...
0
votes
1answer
58 views

Eigenfunctions of a second derivative operator

Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\alpha >0$ is a positive real ...
0
votes
0answers
15 views

wave propagation - Memory variables and weak formulation

Working on the acoustic wave equation (not in terms of displacement $u$ but in displacement potential $X$ s.t. $\rho u = \nabla X$), I'm trying to get a proper viscoacoustic model by the mean of ...
0
votes
1answer
44 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
0
votes
1answer
41 views

spectrum of compact operators

Let $\phi\in\ell^\infty(\mathbb{N})$. For $p\in[1,\infty]$, define $$M_\phi:\ell^p\to\ell^p,\quad f\mapsto\phi f.$$ Use spectral theory to show that, if $M_\phi$ is compact, then $\phi\in c_0$. Here ...
4
votes
1answer
241 views

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
0
votes
1answer
36 views

Where is a mistake in my proof concerning the spectrum of elements of a unital Banach algebra?

I was going to prove: Let $A$ be a unital Banach algebra. Then $$\sigma(a) = \{\tau(a) \mid \tau \in \Omega (A)\}$$ and I started the following argument: Let $\lambda \in \sigma (a)$ and let ...
1
vote
1answer
143 views

Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal ...