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

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Eigenvectors of a Self-adjoint Differential Operator Spans the Domain

Can someone, please, suggest a reference or what steps should I take to prove the following theorem: The set of eigenvectors of a self-adjoint differential operator, defined over a finite ...
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1answer
112 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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1answer
132 views

Continuous functional calculus

Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ ...
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1answer
71 views

Convergence of Fourier series - strange graph in proof

I am reading a text that states the following related to convergence of Fourier series: $$g_K(x) = > ...
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98 views

Global solution for spectral clustering

I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
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93 views

Spectral density function in stationary process

I have the following argument in my Time Series class notes: Let ${u_t}$ be a mean zero covariance stationary process. Define $\gamma(j) = \mathbb{E}u_tu_{t-j}$ and $Y_t = \mu +C(L)u_t$ where ...
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227 views

Spectral theorem of compact operators in Hilbert space

I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof. Theorem. Let $H$ ...
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1answer
106 views

Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 ...
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68 views

Spectrum of the Hill Operator $L(y)= -y''+ v(x) y $

Consider the eigenvalue equation for the Hill operator $$L(y)= -y''+ v(x) y = \lambda y, \quad x\in \mathbb{R},$$ where $v(x)$ is any potential and $\lambda$ is the spectral parameter. If $v(x) ...
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71 views

Spectral theory - a simple application

I'm following Martin Schechter's 'Principles of Functional Analysis' (Second Edition, 2002) and am interested in the spectral theory chapter (chapter six). In particular, I wish to make us of ...
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56 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
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In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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58 views

Determining the spectral density?

Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and ...
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1answer
34 views

The spectrum of $C(K)$ where $K$ is a compact Hausdorff space

Let $K$ be a compact Hausdorff space, what's the spectrum of $f\in C(K)$? I don't know how to start.
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51 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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2answers
99 views

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
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77 views

Spectrum property

Can someone please provide good demonstration for this theorem: Theorem 1.9. Let $A$ be a Banach algebra. If the elements $a, b \in A$ satisfy $ab = ba$, then $\sigma (a + b) ⊆ \sigma (a) + \sigma ...
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2answers
527 views

Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
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259 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
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131 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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Eigenvalues of a connected graph $G$ are greater than or equal to $-1$ iff $G$ is perfect?

Consider $P_G$ as the characteristic polynomial of the adjacency matrix of the connected graph $G$. It is easy to prove that $P_{K_n}(x)=(x-n+1)(x+1)^{n-1}$, so all of the eigenvalues of a perfect ...
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why is the spectrum of the schrödinger operator discrete?

let (M,g) be a compact riemannian manifold. Then the spectrum of the Schrödinger opartor $H=-\Delta +V$ with bounded potential V acting on $L^2(M)$ consists of discrete Eigenvalues ...
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369 views

Spectral radius inequality

Suppose $A,B \in M(n \times n, \mathbb{C})$ or $ A,B \in M(n \times n, \mathbb{R}) $. Under wich hypothesis can I state that: $\rho(AB) \leq \rho(A)\rho(B)$ ?
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Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
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Help computing an integral for Green's function of a discrete Laplacian on a square lattice

I need to calculate the following integral: $$ \int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2 $$ I have tried to use some contour ...
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196 views

Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

in the Wu-sprung model, given a Hamiltonian in one dimension $$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$ we can define the function $ f(x) $ implicitly as $$ f^{-1}(x)= 2\sqrt{\pi} ...
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55 views

Relationship between spectrum and Norm of bounded linear maps .

I am reading the following paper : http://www2.icmc.usp.br/~sma/cadernos/toc9.1/292.pdf In the second paragraph the author introduces a new operator $\|x\|_{T,\epsilon}$ , which i don't really ...
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50 views

Convergence of spectrum along with the convergence of the Operator.

This seems to be very interesting result , ie If operators $\{A_n\}$ in Banach space $B(X)$ and if $A_n \to A$ , $A \in B(X)$in operator norm then $\lambda_n \in \sigma(A_n)$ ie spectrum of $A_n$ ...
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Computing the spectral decomposition for the multiplication operator $f(x) = \frac{1}{1+x^2}$

I am trying to use the spectral theorem for self adjoint operators to decompose the spectrum of the multiplication operator $f(x) = \frac{1}{1+x^2}$ on $L^2(\mathbb{R}).$ This is a problem in Teschl's ...
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Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
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119 views

Spectral radius of an operator .

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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294 views

Eigenvalues integral operator - general case

Let $T$ be an integral operator on $L^2([0,1])$, such that: $$ (Tf)(x) = \int_0^1K(x,y)f(y)dy, $$ with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
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Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
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231 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
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Spectrum of a weakly compact operator

It is well known that the power of a weakly compact operator is compact. Is the spectrum of a weakly compact operator is the same as a compact operator?
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158 views

Invertibility of $I-A$ if the spectral radius of the operator $A$ is less than $1$

I want an explication of the following fact: If the spectral radius of a bounded operator $A$ on a Banach space is less than one, then $I - A$ is invertible.
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135 views

Spectra of a multiplication operator on $C^0(\overline\Omega)$

Let $\Omega \subset \mathbb{C}$ open and bounded, $X = C^0(\overline{\Omega})$ (vector space of continuous complex-valued functions). For an $y \in X$ with $y(\overline{\Omega}) = \{\lambda \in ...
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114 views

Diagonalization discrete laplace operator

How do i diagonalize the discrete laplace-operator $\triangle$ on $l^2(\mathbb{Z})$ (defined by e.g. $\triangle e_k = e_{k-1} + e_{k+1}$, with $e_{k}$ being the canoical basisvectors of ...
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1answer
381 views

Compactness and spectrum of integral operator

Show that the operator $C: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $$Cf(x) = \int_0^x\int_1^tf(s)dsdt$$ is compact and determine its spectrum. Im not sure how to find the spectrum when we are ...
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518 views

Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
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$L^p$ norm of diagonal is $\leq$ Schatten $L^p$ norm

Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove ...
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Proof that the spectrum of the Dirichlet Laplacian is discrete

Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I ...
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178 views

Spectrum of a multiplication compact operator

Let $X=(C([0,1]), \Vert \cdot \Vert_{\infty})$. Determine the spectrum of $$ \begin{split} M \colon & X \to X\\ & u(t) \mapsto \int_0^t h(s)u(s)ds \end{split} $$ where $h \in C([0,1])$ ...
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Borel - Caratheodory Inequality

If $f$ is a complex-valued function analytic on $\{z:\vert z \vert \leqq r \}$, then for $\vert z \vert <r $, $$ \vert f(z) \vert \leqq \frac{2\vert z \vert}{2-\vert z \vert} \sup\{\Re f(w): \vert ...
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1answer
84 views

Spectra of operators

Please help me proof a theorem: If $\mathfrak{U}$ is a complex, commutative Banach algebra with identity and $x\in\mathfrak{U}$, then $$ \sigma(x)=\{\phi(x):\phi \text{ is a homomorphism of } ...
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1answer
176 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
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Properties of the spectrum

Let $\rho$ denote the resolvent of a closed operator and if $\lambda \in \rho(A)$, define $R(\lambda,A) := (\lambda I -A)^{-1}$. If $\mu$ is sufficiently small ...
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265 views

Compute the spectrum for a operator

Find the spectrum of the operator $$ \begin{split} A & \colon C[0,1] \rightarrow C[0,1] \\ & f \mapsto (Af)(x) := f(x) + \int_0^x f(t)dt \end{split} $$ P.S.: I know the spectrum ...
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What's wrong with this spectrum of a “scalar product” in $l^2$?

Let $T\in B(l^2)$ be s.t. $Tx=(\alpha_1 x_1, \alpha_2 x_2, \cdots )$, where the set of all $\alpha$ is dense in $[0,1]$. I've shown that the set of all eigenvalues is $A=(\alpha_j)_1^\infty$. The ...