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

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Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
2
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1answer
56 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
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48 views

Infinite Dimensional Vector Space Equivalence of Positive Matrix

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
3
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1answer
57 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
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24 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
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1answer
54 views

Domain of densely-defined second derivative operator, and its factorization

Let $$-d_x^2: \{f \in L^2[0,1];f \in AC^1[0,1] , f(0)=f(1)\} \rightarrow L^2[0,1]$$ be the second derivative operator. Here $AC^1[0,1]$ is the space of functions whose first derivative is absolutely ...
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2answers
53 views

Decomposition of a positive semidefinite self-adjoint operator?

If I have a positive semi-definite self-adjoint operator $H:D(H) \rightarrow L^2$, is it true that there is always a decomposition $H=A^* A$ available? If this is true, what can we say about the ...
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0answers
40 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
4
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2answers
107 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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0answers
15 views

Characteristic Function of a Spextral Density Function

I am struggling with understanding the link between the Spectral Density Function and the Characteristic Function. In particular, can you find the Characteristic function when only the SPX and ...
1
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1answer
43 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
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1answer
35 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
3
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2answers
103 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
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1answer
101 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
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1answer
26 views

logarithm of projection

I want to prove what's used in the fourth line below the "Proof" section here: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result The statement is: Let $\rho$ be a density operator on a ...
2
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0answers
58 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
2
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1answer
40 views

To What Extent is the Fourier Inversion Theorem Due to the Self-Adjointedness of the Laplacian

I've tried looking this up (I looked at various spectral theorems) but couldn't find anything that talks about the connection between Fourier transforms and the eigenfunctions of the Laplacian (we may ...
3
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1answer
68 views

Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
2
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1answer
30 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
0
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1answer
37 views

Pseudospectrum of an $n\times n$ matrix has at most n connected components

For $\epsilon>0$, the $\epsilon$ pseudospectrum of an $n\times n$ matrix $A$ is given by $\sigma_{\epsilon}(A)=\{z\in \mathbb{C}:\|(z-A)^{-1}\|>\epsilon^{-1}\}$, with the convention that ...
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1answer
62 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
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1answer
88 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
4
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1answer
101 views

Proving the spectral theorem for unbounded self-adjoint operators

Let $A$ be (densely-defined) self-adjoint operator on a (complex) Hilbert space $H$. Then, the Cayley transform $U=(A-i)(A+i)^{-1}$ is a unitary operator, and $(A\pm i)^{-1} \in B(H)$. Using the ...
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56 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
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1answer
13 views

Question about Joint spectral radius.

Given a bounded set $\mathcal A\subset \Bbb R^{n x n}$. The joint spectral radius is given by: $\sigma(\mathcal A)$=$limsup_{m\to\infty}(sup_{A\in\mathcal A^m} \rho(A))$ where $\rho$ is the normal ...
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1answer
21 views

Diagonalisation spectral theorem

For the proof of the spectral theorem for complex numbers I know that the proof follows that, as T is normal then the algebraic and geometric multiplicities coincide. This means that there will be n ...
2
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1answer
79 views

Eigenvalues of polynomials of a matrix and its inverse up to summation by identity

There is a paper that I am reading and the following has been considered without proof: (Suppose $\lambda(.)$ defines the spectrum of a matrix and one can define a random variable on this spectrum say ...
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2answers
51 views

Spectrum of a bounded operator $T$ satisfying $T^n=I$

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, suppose $T\in \mathcal{B}(\mathcal{H})$ is a bounded operator and suppose that $n$ is the smallest natural number so that $T^n=I$. Let ...
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1answer
14 views

A little question about using spectral mapping theorem for polynomials

A question from Kreyszig: Let $X$ be complex Banach space with $T\in B(X,X)$ and a $p$ a polynomial. Show that the equation $p(T)x=y$ has a unique solution $x$ for every $y \in X$ if and only if ...
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1answer
43 views

Spectral radius of an operator equals its norm

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. We know that the spectrum of $A$ is always included in the ball $B(0,|A|)$ and the spectral radious $r(A)$ is the smalest radius such that ...
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46 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
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21 views

Spectrum of the sum on a tensor product?

I have the following problem. Consider the operator $R= H\otimes 1 + 1 \otimes K$ on the tensor product $\mathcal H \otimes \mathcal K$ where $H$ and $K$ are self-adjoint. I know that $R$ has a ...
0
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1answer
54 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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1answer
39 views

A decomposition of Hilbert space via self-adjoint operator

Let $H$ be a complex Hilbert space and $A:H\to H$ self-adjoint. Show that one can decompose $H$ into two $A$-invariant closed subspaces as $H=H_{p} \bigoplus H_{c}$ such that the spectrum of ...
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27 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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10 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
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44 views

Basic Criterion on Selfadjointness

How to prove the following in a neat subjective way: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$
5
votes
2answers
89 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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0answers
38 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
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1answer
80 views

Book: Functional Calculus

Is there a good book that investigates in detail the various kinds of functional calculus? I'm having now some knowledge about unbounded operators and integration but I would like to understand ...
0
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1answer
89 views

Application of eigenvalueproblems for the wave equation

I'm currently searching for a nice little application of an eigenvalueproblem and found the following for acoustics - but one part doesn't make sense for me. Consider the wave equation to find some ...
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1answer
66 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
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1answer
31 views

Why is this subspectrum closed

Let $u: X \to X $ be a compact operator on a Banach space $X$ and let $\lambda \in \mathbb C$ be non zero. We know that $u-\lambda$ is Fredholm and that $X=\mathrm{ker}(u-\lambda)^n \oplus ...
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69 views

Symmetric Operator $\iff$ Real Spectrum

One the one hand not every symmetric operator has real spectrum: $$A\text{ symmetric}\nRightarrow\sigma(A)\text{ real}$$ (In fact this is true only for the self adjoint ones.) Does the converse fail ...
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53 views

Normal Operators: Spectrum vs. Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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votes
2answers
62 views

Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?
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1answer
56 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
2
votes
1answer
42 views

Cayley Transform: well defined?

Why is the Cayley backtransformation well-defined: $$A_U:=\imath(1+U)(1-U)^{-1}$$ In general $1-U$ is not invertible for example for $U=1$.
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2answers
56 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
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61 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...