Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [spectral-theory]

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

0
votes
1answer
23 views

Spectral theorem on simple integral operator

Let $K(x): \mathbb R^d \to \mathbb R^d$, $K \in L^1(\mathbb R^d)$ such that $|K(x)| < M$ We define the integral operator $T$ as such: $\displaystyle(Tf)(x) = \int_{\mathbb R^d}f(y)K(|x-y|)dy$ ...
2
votes
1answer
19 views

Convergence in C* algebra forces spectra to approach limit point's spectrum

Let $A$ be a C* algebra with unity and let $x$ be a normal element and $(x_n) $ a sequence of normal elements converging to $x$. Also let $\Omega$ be a compact neighborhood of $\sigma(x) $. I am ...
-1
votes
0answers
14 views

Estimates for pseudodifferential operator [on hold]

What is the difference between hypoelliptic estimate and subelliptic estimate? Thanks
0
votes
0answers
39 views

Book recommendation request on Spectral theory

Can someone please recommend to me a text that deals on spectral theory from the scratch covering the parts of a spectrum (approximate, point and compression) explicitly. Theorems and properties. ...
1
vote
1answer
13 views

Unbounded operator and generator

Let $\mathcal{H}$ be a Hilbert space. Let $\{U(t)\}_{t \in \mathbb{R}}$ be a strongly continuous unitary group, such that $\forall f \in \mathcal{H}$: $$f(t + \theta) = U(t)f(\theta)U(t)^{-1},\; \...
1
vote
1answer
17 views

inverse of self-adjoint operator is self-adjoint

Given a (unbounded) self-adjoint operator $T: D(T) \subset X \to X$. Assume that $T^{-1}$ exists. Is it true that $T^{-1}$ is self-adjoint? My understanding is that for any $r,s \in X$, there exist ...
3
votes
1answer
85 views

When is $\exp(-iHt)$ well-defined?

If $H$ is a linear operator, what restrictions should be put on $H$ in order for $\exp(-iHt)$ to be well defined? How do you define $\exp(-iHt)$ when $H$ is infinite-dimensional? (If it is possible)
1
vote
1answer
44 views

Spectral analysis of a $2\times 2$ matrix

We have the matrix $$ A=\begin{pmatrix}-\sqrt{3} & 3 \\ 3 &\sqrt{3}\end{pmatrix}. $$ I want to find the spectral analysis of that matrix, i.e., write $A$ as a linear combination of ...
5
votes
0answers
90 views
+50

Prove that a kernel operator has no eigenvalues

Good evening! I'm just popping here for a quick question. I'm just starting to work on kernel operators, from $L^2(\mathbb{R}_+)$ to itself, ie: $f \mapsto \left(x \mapsto \displaystyle\int_{\mathbb{...
1
vote
0answers
14 views

On the connections between the spectral theorem and functional calculus for normal operators.

I am interested on some connections between the spectral theorem for normal operators and a corresponding functional calculus which can be assigned to them. I'll begin by recalling these two results, ...
2
votes
0answers
22 views

Obtaining directional derivatives from the gradient of the projection onto the positive semidefinite cone.

I am currently having a hard time with the notation for derivatives of spectral functions. In 2006, Malick and Sendov in (DOI: 10.1007/s11228-005-0005-1) have derived an explicit form for the second ...
0
votes
2answers
30 views

Operator with given spectrum which is not projector.

I'm stuck in making an example of such operator $A$, that spectrum of $A$ is $\{0,1\}$, but A is not a projector. Could you give me such example, please
2
votes
1answer
26 views

Spectrum of compact operator on infinite dimensional Hilbert space is countably infinite

So, I've been working on this question for a while now and cannot get it right. I have a compact, self adjoint, positive definite operator $K$ on infinite dimensional Hilbert space. I know that $0\in ...
1
vote
1answer
15 views

Spectrum of an operator $\sigma(L)$

How can I find $\sigma(L)$, $\sigma_{p}(L)$ , $\sigma_{c}(L)$ i , $\sigma_{r}(L)$ for operator $L(x_{1},x_{2},\ldots)=(x_{2},\frac{1}{2}x_{3},\frac{1}{3}x_{4},\ldots)$ ?
0
votes
1answer
35 views

I want to find an example where the Spectrum is equal to the Continuous Spectrum in C[0,1]; $\sigma_c(A)=\sigma(A)$

I tried to find this example but the condition $\overline{\operatorname{range}(\lambda I -A)}=C[0,1]$ is too hard to prove. Anyone could help me?
1
vote
0answers
24 views

Basic measures and $L^\infty$ functional calculus

I'm recently studying some topics related to spectral theory and I found out that one can extend the Borel functional calculus (that is a -* omomorphism from Borel complex valued bounded functions to ...
1
vote
1answer
46 views

The spectral decomposition of skew symmetric matrix

I have been studying the spectral decomposition of the matrices and figured out that it works for symmetric matrices but it wont work for the skew symmetric ones well, the sign of the final matrix is ...
1
vote
0answers
24 views

Spectral Theory of Markov Jump Processes (continuous-time)

I stumbled over the eigenvalue problem while analysing an infinite-dimensional jump process. The state space I am working with is $$ \mathbb{N}^{<\infty}=\bigcup_{k=1}^{\infty}\mathbb{N}^k $$ i.e. ...
1
vote
1answer
32 views

Does the spectrum at a point vary continuously in this case?

Let $A$ be a C$^{*}$-algebra. Let $\hat{A}$ denote the set of all irreducible representations of $A$. Suppose $\pi\in\hat{A}$ has the following property: for all $a\in A$, the map from $\hat{A}\to\...
1
vote
1answer
33 views

Restriction of a compact operator on a finite-dimensional subspace

I have a self-adjoint compact operator $\Gamma : L^2[0,1] \to L^2[0,1]$ with positive eigenvalues $\lambda_j$, which of course tend to zero,and a general finite dimensional linear subspace $S \subset ...
0
votes
1answer
12 views

Computations of a counterexample in order to check that the sum and product of closed operators are not always closed

While I was studying functional analysis I found in the script the following counterexample: Let $X = l^1$ and consider the linear operator $$ (Ax)_n\left\{ \begin{array}{ll} n x_{n-1} ...
3
votes
1answer
44 views

Inverse of (i*I+A) for self adjoint operator

If we have that $A: H \rightarrow H$ is a bounded self-adjoint operator on a Hilbert space, then the spectrum of $A$ is entirely real, i.e. $\sigma(A) \subseteq \mathbb{R}$. Hence we know that $i$ is ...
3
votes
0answers
29 views

Why the space have to be complex in the spectral theorem for bounded self-adjoint operators?

In the spectral theorem for compact self-adjoint linear operators $T:H\to H$ (as stated in Conway's book), the Hilbert space $H$ can be real or complex. However, in the spectral theorem for bounded ...
2
votes
0answers
29 views

Find the Spectra of this Bounded Linear Operator

Let $X$ denote the space $L^{1}(\mathbb{R})$ of all (equivalence classes of) Lebesgue integrable functions $f:\mathbb{R} \to \mathbb{C}$ with the norm $||f||_{1} = \int_{\mathbb{R}}|f(t)|dt$. Let $T \...
1
vote
1answer
21 views

Does the limit of this sequence of operators have infinitely many eigenvalues?

Suppose that I have a sequence of compact, injective operators $\{T_\delta\}_{\delta>0}$ on a Hilbert space $H$ such that each operator $T_\delta$ has infinitely many eigenvalues. My question is ...
0
votes
0answers
6 views

Dirichlet Spectrum of Laplacian In 2D Annulus

What are the eigenvalues and eigenfunctions of the Euclidean Laplacian for the annulus in 2D bound by radii r_1 < 1 < r_2 if we consider zero Dirichlet boundary conditions? I’m aware that this ...
1
vote
0answers
40 views

Spectrum of the difference of two commuting operators

I am looking for a way to prove that for two commuting, linear and bounded operators $A,B$ acting on a Banach space $X$ $$\sigma(A-B)\subset\sigma(A)-\sigma(B)$$ I have already found a proof that "...
0
votes
0answers
19 views

Abstract Dirac operator: domain left invariant by involution?

I'm willing to prove here that the Dirac operator: $H_{0} = -ic\alpha.\nabla + \beta mc^{2}$, defined on $H^{1}(\mathbb{R}^{3}, \mathbb{C}^{4})$, is an abstract one. By definition, an abstract Dirac ...
2
votes
0answers
31 views

Spectral Theorem: Realization of a direct sum of $L^2$ spaces as a single $L^2$ space

The following is motivated by an attempt to understand the Spectral Theorem for Bounded operators on a none separable Hilbert space. One version of the theorem states that for a bounded (say normal) ...
1
vote
1answer
35 views

Laplacian commutes with covering maps

I believe that this should hold true: Result. Let $\pi: M \to N$ be a Riemannian covering ($M$,$N$ closed). Then $\Delta_M \circ\pi^* = \pi^* \circ \Delta_N$. This should be true, but I could not ...
1
vote
1answer
33 views

Spectrum of operator $(Tx)_{i}=\sum_{j=0}^{n}\alpha_{j}x_{i+j}$

Note: Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you. Let $\alpha_{0},\ldots,\alpha_{n}\in\mathbb{C}$ be given. Compute the spectrum of $T:\...
0
votes
1answer
19 views

Why is the spectrum of the hamiltonian for an infinite square well just a point spectrum?

Consider the Hamiltonian $H = -\Delta + V$ where $V$ is the potential conrresponding to an infinite square well: $$V(x) = \begin{cases}0,&\text{if } 0, \leq x \leq L;\\\infty,&\text{otherwise}...
0
votes
0answers
60 views

A question on spectral decomposition

I'm working on a quantum information processing question but my question is purely maths based, it regards the spectral decomposition of the following matrix : Note: I believe (though I may be wrong ...
6
votes
1answer
31 views

Hyponormal operator and approximate spectrum

Let $H$ be a complex Hilbert space. It is well known that if $T:H \to H$ is a normal operator, then $$\sigma(T)=\sigma_{ap}(T),$$ where $\sigma_{ap}(T)$ is defined as: $\lambda \in\sigma_{ap}(T)$ ...
0
votes
2answers
42 views

Continuous and residual spectrum

Note: Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you. I am having trouble understanding and finding the continuous and residual spectrum. I ...
0
votes
1answer
44 views

Spectrum of product of continuous linear operators

Note: Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you. Show that if $S,T\in\mathcal{L}(X)$, where $X$ is a Banach space over $\mathbb{C}$. ...
1
vote
1answer
35 views

what's spectral axiom

I encounter a proposition in an article: For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$ ...
0
votes
0answers
32 views

How to prove Neumann series doesnt converge when spectral radius > 1?

For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$ If $r(T)>1$, I can show that series ...
0
votes
0answers
21 views

There is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$

Could someone give me a suggestion to solve the following problem problem? PROBLEM. Let $T : \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ and $\beta$ a basis of $\mathbb{R}^{3}$ such that $$ \left[ ...
2
votes
0answers
19 views

Proving that two variational problems are equivalent

Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \...
4
votes
1answer
77 views

Point spectrum of an integral operator

Let we have $$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy$$ so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
2
votes
1answer
44 views

Computation of the complex roots of the Laplace transform of a function?

I have a function $f \in L^1(\mathbb{R}_+, \mathbb{R})$ with Laplace transform $$ \forall \Re(z) \geq 0,~~ \hat{f}(z) := \int_{\mathbb{R}_+} { f(t) e^{-zt } dt}.$$ I know explicitly the expression of ...
0
votes
1answer
34 views

Relation between Schatten-$p$-norm and $l^p$ norm of operator matrix

Let $\mathcal H$ be a separable Hilbert space and let $(e_i)$ be some orthonormal basis. Let $K$ be a compact operator on $\mathcal H$ with matrix elements $K_{ij}=\langle K e_i,e_j\rangle$. My goal ...
1
vote
0answers
24 views

Given two commuting projection-valued measures, is there a product measure?

Let $H$ be a Hilbert space. Let $(\Omega_1,\mathcal A_1)$ and $(\Omega_2,\mathcal A_2)$ be measurable spaces. Let $$P_1: \mathcal A_1\to\mathfrak L(H), \quad P_2: \mathcal A_2\to\mathfrak L(H)$$ be ...
0
votes
1answer
27 views

Quadratic Forms Orthogonal Diagonalization Existence

Why does one assume that the eigenbasis for a quadratic form is orthogonal, hence orthogonal diagonalization. I understand that for hermitian and unitary maps one can show by spectral theorem an ...
0
votes
1answer
26 views

How do we derive the spectral projector associated with a simple eigenvalue?

Result 7.2.12 of Meyer's Matrix Analysis and Applied Linear Algebra gives the following: If $x$ and $y^*$ are respective right and left eigenvectors of a matrix $A$ associated with a simple ...
0
votes
1answer
42 views

Trouble understand proof of spectral theorem

I'm reading through this proof of the real spectral theorem. I don't understand the last line of "lucky fact 2" - why must $\overrightarrow{u}$ have been listed in the $v_{i}$?
0
votes
0answers
28 views

Periodic boundary conditions, general dimension, sets and spectral properties of $-\Delta$ - reference recommendation

Let's consider the eigenvalue problem $-\Delta u = \lambda u$ on the interval $[0,1]$ with periodic boundary conditions: $u(0)=u(1),$ $\frac{du}{dx}(0) = \frac{du}{dx}(1).$ Similar conditions could be ...
0
votes
0answers
44 views

Is there a way to concretely see cyclic sub-representations of the “Euclidean group” on $\mathbb Z$?

Let $\mu$ be a finite Borel measure on $S^1$. We have an action of $\mathbb Z$ on $L^2(S^1, \mu)$ defined by $n\cdot \varphi = e^{2\pi i n}\varphi$. The following is a standard theorem in functional ...
1
vote
0answers
25 views

Definition of spectral measures corresponding to koopman operator

Let $U$ be the koopman operator on $L^2(X,\mu)$ where $(X,T,\mu)$ is an MPT. We characterize the spectral measure by its fourier coefficients using bochner's theorem as $$\hat{\sigma_{f,g}}(-k) = \...