Questions tagged [spectral-theory]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
19 views

Spectrum of 'separable' operator over measure spaces

Given $\sigma$-finite measure spaces $(X,\mu)$ and $(Y,\nu)$, can we say something about the spectra of operators on $A\in L^2(X\times Y; \mu\times \nu)$, if $A$ has a seprable sturcture? Something ...
user avatar
  • 5,344
0 votes
0 answers
27 views

Good source to study the Laplace transform.

I am studying the theory of semigroups and its links with the spectral theory and the Laplace transform turns out to be the intermediary between the two. Any suggestions for good sources?
user avatar
0 votes
0 answers
28 views

maximal eigenvalue of self-adjoint operator is non-degenerate

I want some help in this one, if someone can prove or disprove it: "If $T$ is a compact, self-adjoint operator with positive spectral gap, then $||T||_2$ is always an eigenvalue and the ...
user avatar
  • 101
0 votes
0 answers
52 views

Fractional powers of the Laplacian

I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for $$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}....
user avatar
  • 121
2 votes
1 answer
41 views

Connectedness of Toeplitz operator spectrums

Im working through the following theorem, and I don't understand the explanation. Theorem: Let $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ be the Toeplitz operator with $T \subset \mathbb{C}$ the ...
user avatar
4 votes
2 answers
67 views

Continuity of polar decomposition

This question is about a step in the proof of this answer, which is not directly clear to me. Consider the following scenario: $H$ is a Hilbert space, $A\in GL(H)$ a positive self-adjoint operator. We ...
user avatar
  • 2,031
5 votes
1 answer
62 views

If an element in a Banach algebra is anihilated by an analytic function then it must be algebraic.

Let $A$ be a Banach algebra, let $a\in A$ and suppose $f(a)=0$, where $f$ is an analytic function defined on an open set $U$ containing $\sigma(a)$. Prove that $a$ is algebraic in the sense that $p(a)...
user avatar
  • 13.9k
0 votes
0 answers
24 views

Systems of differential equations and spectral theory.

Say that we have the system of differential equations in matrix formulation: $$ \begin{bmatrix} C(t) \\ C_p(t) \end{bmatrix}' = \begin{bmatrix} -k_{cp}-k_{ce} & k_{pc} \\ k_{cp} &...
user avatar
  • 133
1 vote
0 answers
62 views

Spectral theory and and a problem in $\ell^{2}$

Be $\boldsymbol{X}=\ell^{2}$ and operators $A, B: \ell^{2} \rightarrow \ell^{2}$ by $$ A x:=\left(\xi_{2}, \xi_{3}, \ldots\right), \quad B x:=\left(0, \xi_{1}, \xi_{2}, \xi_{3}, \ldots\right) $$ to $x=...
user avatar
3 votes
1 answer
39 views

Normal operator $T$, $\sigma(T) \subset \{0,1\} \Rightarrow$ $T$ is an orthogonal projection

Let $T$ be a bounded normal operator acting from a Hilbert space $H$ to itself. I must check that if the spectrum of $T$ lies in the set defined in the title, then $T$ is an othogonal projection. I ...
user avatar
  • 135
0 votes
2 answers
58 views

If $\lambda$ is in the continous spectrum of $L$, then the range of $L-\lambda$ can not be closed?

Let the continuous spectrum of a densely defined linear operator $L$ over a Separable Hilbert space, be defined as the set of all $\lambda \in \mathbb C$ such that: (i) $L-\lambda$ is injective, (ii) ...
user avatar
  • 61
0 votes
0 answers
7 views

Difference between contour integral operator and trace operator in spectral analysis

The trace operator can be used to trace the boundary of a subspace of a spectrum. Some methods also use the contour integral in spectral analysis. My question is then: is the contour integral used to ...
user avatar
0 votes
0 answers
15 views

Question about elliptic boundary condition

I'm a little confused with showing that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. Could give please give me some help with the details? Thanks in ...
user avatar
0 votes
0 answers
26 views

Eigenvalues of graph laplacian question

I need some guidance on solving the following question I am stuck on: Let L be a graph Laplacian matrix of a graph G = (V, E) with n vertices. Denote the eigenpairs of L by {λi,ψi} for every i=1..n, ...
user avatar
  • 11
1 vote
1 answer
38 views

Showing 1 cannot be an eigenvalue after rotating an operator

I have a compact self-adjoint positive integral operator $Q:L^2(0, \infty) \to L^2(0, \infty)$ with operator norm $\| Q\| =1$. By the assumptions, we know $1$ is an eigenvalue of $Q$. Let $y\ne 0$ and ...
user avatar
1 vote
2 answers
35 views

Is the spectrum of $AB+C$ the same as $BA+C$?

It is known that for matrices $A$ and $B$, $AB$ and $BA$ are isospectral. However, I am not sure if this holds for a when these terms are in a sum, i.e. if $AB+C$ is isospectral to $BA+C$. My ...
user avatar
  • 1,357
0 votes
0 answers
28 views

Check self adjointness of element

Let $A$ be a unital $C^*$ algebra, $u$ an unitary element such that $\sigma(u)\neq\mathbb{T}$ (the spectrum). Consider the usual complex $\log$ function (with the usual branch). Under these ...
user avatar
  • 566
0 votes
1 answer
51 views

If A is symmetric and $A^2 = A$, show it is a projection matrix.

I assumed showing $A=QQ^T$ was enough to say it is a projection matrix, but I think it's not. I showed this by saying $A=QDQ^{-1}$ (D is diagonal) and knowing $Q^T = Q^{-1}$ for orthogonal matrices, ...
user avatar
1 vote
0 answers
40 views

Spectral properties of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)}~~~~~~~~~~~~~~~...
user avatar
0 votes
1 answer
32 views

Is $r \in \text{Rad}(A)$?

Let $u$ be an element of a Banach algebra $A$ such that $u^2-u \in \text{Rad}(A)$. I am trying to show that there exists a projection in $A$ which is equal to $u$ modulo the radical. We have that $\...
user avatar
  • 3,869
5 votes
0 answers
46 views

Is every bounded linear functional on a subspace of a Hilbert space given by a function?

This question comes from the Limiting Absorption Principle (LAP). I want to obtain the most general statement possible, so I proceed as follows. Let $M$ be a topological space, $(H, \langle \cdot, \...
user avatar
  • 51
1 vote
0 answers
23 views

Logarithm of bilateral shift $-i\log(U)$ acting on basis elements of $\ell^{2}(\mathbb{Z})$

Consider $\ell^2(\mathbb{Z})$ with standard basis $\{e_n\}_{n\in\mathbb{Z}}$ and the bilateral (right)-shift: $U(e_n)=e_{n+1}$. The spectrum of $U$ is the unit circle in the complex plane $\mathbb{C}$....
user avatar
0 votes
3 answers
45 views

What is the number of linear independent eigenvectors of a complex matrix when the characteristic and minimal polynomials are the same? [closed]

Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$ such that $$p_{A}(t) = m_{A}(t) = (t - \...
user avatar
  • 553
6 votes
1 answer
68 views

Example of an ergodic transformation with some properties in the spectrum of the Koopman operator

Let $(X,\mathcal{B},\mu)$ a probability space, for simplicity we can assume $X$ a metric space. Let $T: (X,\mathcal{B},\mu) \to (X,\mathcal{B},\mu) $ be an invertible, measure-preserving ...
user avatar
  • 2,218
4 votes
0 answers
37 views

How to derive the Sommerfeld radiation condition from the resolvent?

Suppose the resolvent $R_0(\zeta)$ for an operator $P_0(\zeta)$ is known to satisfy a limiting absorption principle at $\lambda \in \Bbb R$. For some perturbation $P(\zeta)$ of $P_0(\zeta)$ with $V (\...
user avatar
0 votes
0 answers
33 views

Size of essential spectrum if $T-\lambda$ is not injective for all $\lambda$ in the essential spectrum.

Let $B$ be some Banach space and let $T:B \to B$ be linear and bounded. I write $\sigma_e$ for the essential spectrum, i.e. the set of $\lambda \in \mathbb{C}$ s.t. $T-\lambda$ is not Fredholm. The ...
user avatar
  • 407
1 vote
0 answers
28 views

Eigenvalues ​decreasing to zero

Let $(H, (\cdot,\cdot))$ be infinit dimensional separable Hilbert space. Also considerer $T : H \rightarrow H$ a non-null compact, self-adjoint operator such that $$ (T(v),v) \geq 0, \forall v \in H.\...
user avatar
1 vote
1 answer
129 views

Reverse spectral theorem proof

Let $T$ be a linear transformation in a finite inner product space $V$ , let $l_1,...l_k$ be different scalars , and let $P_1 \not=0,....,P_k \not=0$ linear transformations in $V$ that satisfies the ...
user avatar
  • 823
0 votes
1 answer
33 views

How is the spectral decomposition of a skew-symmetric matrix $A$ related to that of $iA$?

Any skew-symmetric matrix can be written as $A=UQU^\dagger$ where $U$ is unitary and \begin{align*} Q=\begin{bmatrix} 0 & \lambda_1 & \\ -\lambda_1 & 0 & \\ & & ...
user avatar
  • 171
1 vote
1 answer
38 views

I don't understand our proof for "The spectral radius of a normal operator $A$ is equal to the norm of $A$", it's different to what I've found here

Our proof is a little bit different than the proofs I've found here on this forum. I understand the rest of the proof as it's just using the Gelfand-Beurling formula, but the part I don't understand ...
user avatar
  • 319
0 votes
0 answers
26 views

Point spectrum of restriction

Let $T:D(T) \to H$ be closed unbounded linear operator on a Hilbert space $H$. Let $E \subset H$ be a closed subspace such that $T(D(T) \cap E) \subset E$. Consider the operator $S:D(S) \to E$ ...
user avatar
0 votes
0 answers
39 views

Operator on Hilbert space restricted to orthogonal complement

Let $T$ be a compact self adjoint operator on a Hilbert space $H$. If $(e_n)_n$ is an orthonormal system of eigenvectors then setting $H_0= \{ z : \langle z , e_n \rangle=0$ forall $n$ $\}$, we have $...
user avatar
3 votes
1 answer
68 views

Weak convergence of Reversible and ergodic Markov chain on uncountable state space,

I think this may be a trivial question. But I could not find a proper reference due to my lack of knowledge. Suppose I have a Markov chain $X=(X_n)_{n\geq 1}$ on the uncountable (but compact!) state ...
user avatar
1 vote
1 answer
29 views

Spectrum of a compact operator - proof clarification

I'm reading through the book Linear Analysis by Bollobás. I'm having a bit of trouble understanding part of the proof of theorem 7 in chapter 13, which states: Let $T$ be a compact operator and ...
user avatar
  • 4,624
0 votes
0 answers
27 views

Question about Dirac operator

Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that $$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$ for $\...
user avatar
6 votes
1 answer
197 views

Compact integral operator on $H^1(\mathbb{R})$

Consider the operator $$ {\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy. $$ Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself? To give some context to the problem ...
user avatar
1 vote
1 answer
36 views

Checking a possible corollary of Gelfand's formula

I have recently come across Gelfand's spectral formula for matrices, which Wikipedia gives as follows For any square matrix $A$ and matrix norm, we have $\lim_{k \to \infty} \lVert A^k \rVert ^{\frac{...
user avatar
  • 3,823
1 vote
1 answer
25 views

Integrating unbounded functions w.r.t. a projection-valued measure

I was looking at Frederic Schuller's lectures on quantum theory and there, when defining integrals of unbounded functions with respect to a projection-valued measure, it is left as an exercise to ...
user avatar
  • 225
0 votes
2 answers
52 views

Why does the problem $A u = \lambda u$ bother us in Spectral Theory? Why are we interested in the inverse of $(A - \lambda I)$, and not simply of $A$?

I simply don't see why it should be of our interest to check the eigenvalues of an operator and to find the resolvent of an operator. An eigenvalue here just shows for which $\lambda$ and for which ...
user avatar
  • 459
0 votes
0 answers
21 views

Distribution of eigenvalues of a stochastic matrix

I am given a stochastic matrix. Frobenius-Perron theory tells that $1$ is always an eigenvalue, which is rather obvious. Secondly, it is known that the spectral radius to me is $1$. What else can be ...
user avatar
  • 23
1 vote
1 answer
67 views

Find an expression for $(\lambda\textbf{1}-a)^{-1}$.

Let $A$ be a unital Banach algebra, and suppose $a\in A$ has the property that $a^2=1$, but that $a\neq \textbf{1}$ and $a\neq -\textbf{1}$. By the spectral mapping theorem, $\sigma(a^2) = \{\lambda^2 ...
user avatar
  • 3,869
0 votes
0 answers
31 views

Complement of spectrum of a invertible self-adjoint element has no bounded components

Let $A$ be a unital $C^*$-subalgebra of a $C^*$-algebra $B$ and $x$ be an invertible element of $B$. Let $y$ be an element of $B$ defined by $y=x^*x$. Now $x$ is invertible implies $y$ is invertible ...
user avatar
  • 1,551
1 vote
1 answer
24 views

Compactness of group of unitaries that commute with density operator

I study the paper "Doplicher and Longo: Standard and split inclusions of von Neumann algebras" and have a question about the proof of Lemma 3.2. Let $\omega$ be a faithful normal state on a ...
user avatar
  • 186
2 votes
1 answer
66 views

Spectrum of positive linear operators defined for every $L^p$ space.

Consider the probability space $([0,1], \mathcal B([0,1]), \lambda(\mathrm{d} x) )$, where $\mathcal B([0,1])$ is the Borel $\sigma$-algebra and $\lambda$ the Lebesgue measgure on $[0,1]$. Let $P,T:L^...
user avatar
5 votes
1 answer
84 views

Prerequisites for Bourbaki spectral theory (French)

I am considering reading Bourbaki's text Théories spectrales (Springer, French, 2nd Ed.) and was wondering what the prerequisites are. Currently I am familiar with their texts on the Theory of Sets, ...
user avatar
  • 61
0 votes
1 answer
21 views

Explicit expression for Fredholm determinant.

Let $A$ be a rank $2$-matrix, i.e. there are vectors $v_1,v_2,w_1,w_2$ such that $$Ax = \langle v_1,x\rangle v_2 + \langle w_1,x \rangle w_2.$$ I wonder if there is an explicit expression for the ...
user avatar
  • 169
2 votes
1 answer
36 views

How can the spectrum of $A = \frac{d}{dx}$ in $C(\mathbb{R})$ be equal to $\mathbb{R}$, as for every $\lambda$ there exists a resolvent $R(\lambda)g$?

Let's now find the eigenvalues of $A$ (which are the spectrum): $$A f = \lambda f$$ $$f' = \lambda f$$ $$f' - \lambda f = 0$$ $$\Rightarrow \quad f(x) = C \cdot e^{\lambda x} \quad \Rightarrow \quad \...
user avatar
  • 459
2 votes
2 answers
95 views

Good books and lecture notes to learn pseudo-differential operators and spectral theory

I am looking for a list of good books and lecture notes to learn pseudo-differential operators and spectral theory (for infinite dimensions.) I am familiar with introductory functional analysis, ...
user avatar
2 votes
0 answers
17 views

Maximum diameter of convex cone of solutions

I am trying to find an upper bound for the maximum distance between 2 vectors that satisfy $$x, y \in \{s|As \geq 0 \text{ and } \lvert \lvert s \rvert \rvert_2 \leq L\}$$. What is the maximum ...
user avatar
1 vote
2 answers
78 views

Continuous spectrum of an integral operator

I understand that the continuous spectrum of an operator are the $\lambda's$ such that $(\lambda-T)$ is injective but ran$(\lambda-T)$ isn't dense in the image. But i can't properly calculate it for a ...
user avatar

1
2 3 4 5
55