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

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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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203 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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149 views

The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$? Edit: What if the matrix has positive ...
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270 views

Approximate Point Spectrum is subset of Spectrum

I'm trying to prove that if $\lambda$ an approximate eigenvalue of $T$ then $\lambda \in \sigma(T)$, but I can't work out how to do it. Could someone give me a hint, or point me in the direction of a ...
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52 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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68 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 ...
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43 views

How to find the spectrum $\sigma_p(P)$

How to find the spectrum $\sigma_p(P)$: Let $P:H\rightarrow H$ be an orthoprojection, $P\neq 0, P\neq I$. could you please help
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Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
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Why is this spectrum the closure?

I've started to learn about spectral theory and I'm looking at some examples. The spectrum is defined as $$ \sigma(a) = \{\lambda \in \mathbb C | a-1\lambda \text{ is not invertible} \}$$ If $A= ...
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52 views

Resolvent properties

Suppose that $A$ is a $n \times n$ matrix with $n$ different eigenvalues $\lambda_k.$ Corresponding eigenvectors are denoted as $x_k$, $x_k^Tx_k =1.$ Now $A=X\Lambda X^{-1}$. Denote $Q=X^{-1}$. ...
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74 views

Spectrum of a product

Let $A$ be a unital $C^{*}$-algebra. I am trying to show that if $a,b\in A$ are positive elements, then the spectrum of $ab$ is contained in the positive real numbers. I know that in the commutative ...
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Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
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66 views

Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
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59 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
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224 views

$uu^T$ is the standard matrix for the orthogonal projection of $\mathbb{R}^n$ on the subspace spanned by $u$

Let $A$ be an $n\times n$ symmetric matrix, and $u$ an eigenvector of $A$. Why is it true that $\forall x\in\mathbb{R}^n$, $x^{T}uu^{T}(I-uu^T)x=0$? If I'm able to show this is true then I can show ...
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144 views

Representation of a bilinear form on an Hilbert space

Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated. 1) There exists a symmetric ...
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real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
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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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356 views

Finding Spectral Radius of Matrix

Find the Spectral Radius of $A=$ $\mbox{} \left[ \begin{array}{cc} 1 & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & -c & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} ...
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668 views

spectrum of right shift operator on $\ell^2(\mathbb{Z})$

Consider the right shift operator on $\ell^2(\mathbb{Z})$. Is there a way of calculating (well, showing what it is since I already know it's $z$ s.t $|z| = 1$) its spectrum without reference to it ...
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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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83 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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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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Numerical Range vs Spectrum

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. Prove that for normal operators the spectrum is ...
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26 views

Resolvent of the differentiation operator on the torus

Suppose we have the space $L^2(\mathbb T)$, that is, the space of periodic functions that are in $L^2$. Let $L^0$ be the operator of differentiation, i.e $L^0 f = f'$ where the domain of $L^0$ is the ...
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65 views

Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
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156 views

How to use Parseval' identity( Plancherel)? [duplicate]

(May be this is very basic question for MO) Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix ...
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Spectrum of perturbed operator's

Let $G$ be a normal operator with compact resolvent acting on a Hilbert space $H$ such that $\ker G \neq \{0\}$. If $P$ denotes the orthogonal projection onto $\ker G$, and if $\{\lambda_n\}$ are ...
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If $\lambda$ is isolated in $\sigma(u)$, then $E(\left\{\lambda\right\})(H)=\ker(u-\lambda)$.

This is a Question 2.11 from Murphy's book: C$^*$-algebras and Operator Theory: Let $H$ be a Hilbert space. Let $u\in B(H)$ be a normal operator with spectral resolution of the identity $E$. (a) ...
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Projections in spectral decomposition.

In my quantum mechanics book the spectral decomposition of operator $A$ is given as $A=\sum\limits_j\lambda_jP_j$ where $\lambda_j$ are the eigenvalues of matrix $A$ and $P_j$ is the orthogonal ...
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Perturbation of a matrix with negative eigenvalues

Let $A$ be a square matrix with all eigenvalues negative. What is the relationship between the $\lambda_\max$ of perturbed matrix $A + X$ and the norm of the perturbation $\|X\|$? PS: I know that the ...
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80 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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A transformation recipe for functional calculus of a self-adjoint operator?

Consider a self-adjoint operator $\operatorname{A}$ on a Hilbertspace $\mathcal{A}$ and its spectral decomposition according to the spectral theorem: $$A = \int_{\mathbb{R}} \lambda \;dP_\lambda$$ ...
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139 views

Resolvent R(1) of the Laplace operator not compact

I want to show that $$R_\Delta(1):=(1-\Delta)^{-1} $$ is not compact in $\mathbb{R}^3$. I have found that for $\chi_{B}$ being the characteristic function for a set $B\subset\mathbb{R}^n$, ...
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spectum of self adjoint operators

Let $H$ be an Hilbert space and $S = \displaystyle{ \sum_{i=1}^nS_i}$ where $S_i$ (i=1...n) is self adjoint with compact resolvent . is it true that the spectrum of $S$ is the sum of the spectra of ...
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462 views

The eigenvectors of a matrix and its transpose that correspond to the same eigenvalue are not orthogonal

Spent hours trying to prove this after encountering it in Lax's discussion of the spectral theorem, but no luck. Here's the problem (it is Theorem 18 in Lax 2ed, Chapter 6): A mapping $A$ has ...
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Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
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Eigenvalues of power of matrices

How come if $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$? And is its multiplicity necessarily the same?
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Eigenvectors orthogonal to $j$

I'm studying the proof of the following statement: $Spec(K_n) = (n-1)^1(-1)^{n-1}$ At some point I have: By the Spectral Theorem, when looking for eigenvectors $v$ we can assume they are ...
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57 views

The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Suppose $x$ is invertible in the unital Banach algebra $A$. How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
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114 views

Finding the spectrum of the Schrodinger operator

Let $H(f) = -f'' + V(x) f$ be the Schrodinger operator on $\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator $H$ in $L^2(\mathbb R)$ for various choices of $V$. In ...
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Reading a DFT plot - did I get this right?

I am simulating the evolution of a liquid film through the solution of a 4th order nonlinear partial differential equations. Of late, I began experimenting with DFT of the result that I have. My ...
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127 views

Integral with spectral decomposition

Let $A:H\longrightarrow H$ be a self-adjoint operator, where H is an Hilbert space. Let $(E_{\lambda})_{\lambda}$ be the spectral decomposition of $A$ and $\lambda_0$ a regular value of A with finite ...
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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 ...
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37 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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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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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 ...
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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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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 ...