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

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Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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spectrum of the right shift operator

Here is the question: Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$? Here is what I thought: If one wants to prove that the operator ...
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Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
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Joint spectral radius of $\sigma( \mathcal A)$ and $\rho(A) < 1 \forall A \in \mathcal A$

Given $\mathcal A \subset R^{n \times n}$. The joint spectral radius is by: $$\sigma( \mathcal A) = \limsup_{m \rightarrow \infty}\sup_{A \in \mathcal A^m}\rho(A),$$ where $\rho$ is the normal ...
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Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
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Lower bound for the norm of the resolvent

I need to prove next statement (I want to do it for general case) $\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}$ I think it could be like this let $a\in \sigma(A) z ...
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What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
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Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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Spectrum of left shift operator $L\in B(H)$

Let $H$ be a Hilbert space and $L$ the left shift operator $L\in B(H)$: $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$. I computed the spectrum could someone please tell me if this is right? My work: ...
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Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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Banach Algebra spectral theory [closed]

Let $(\Omega, \mu)$ be a measure space. Show that the linear span of the idempotents is dense in $L_\infty(\Omega, \mu)$.
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Difference between an eigenvalue and a spectral value

What is the difference in the definition of a spectral value and an eigenvalue. My notes from functional analysis says $\lambda$ is an eigenvalue of an operator $A$ if $\,\exists \, x \in ...
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Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
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Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
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Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
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Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
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Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
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are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$th largest eigenvalue of $AB$, in terms of the ...
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Spectral theorem question

I am trying to understand how to develop the spectral measure of a bounded self-adjoint operator on a Hilbert space. For every continuous function on its spectrum, $f: C(\sigma(A)) \to \mathbb{C}$, ...
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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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Sturm-Liouville problem and periodic boundary conditions

I was wondering about this: I know that if a 1-d Sturm-Liouville operator is limit circle or limit point then the eigenvalues are simple ( so no degenerated spectrum). But in the case of periodic ...
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Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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Relation between weighted outer product and unweighted outer product of a matrix

I know this may be quite straightforward, but I don't know how to prove it because it was quite a while since I took undergraduate linear algebra classes. Assume that an $n\times n$ real symmetric ...
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Computation the different spectrums of an operator

Let $X:=C^0([0,2],\mathbb C)$,$\phi\in X$ and $T\in L(X)$ defined as: $$(Tf)(t):=\phi(t)f(t),t\in [0,2]$$ Compute: $\sigma_p(T),\sigma_c(T),\sigma_r(T),\sigma(T)$ and $\rho(T)$ I am quite new to ...
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Can an operator have spectrum consisting of just one point?

Could it happen that an operator has a spectrum consisting of just one point? Sorry if this is really short. Was just pondering to myself random questions and this one came up. The good news is ...
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Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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Are self-adjoint / Hermitian operators necessarily orthogonal / unitary?

I feel like self-adjoint / Hermitian operators are the "best" operators, since an operator that is self-adjoint can be orthogonally diagonalized, according to the Spectral Theorem (over the complex ...
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Spectral Theory of an operator

If we define the spectrum of a bounded linear operator $T$ by $$\sigma(T)=\{\lambda\in \mathbb C:\ T-\lambda I \ \text{ has no inverse} \}.$$ What about $\sigma(T^{-1})$?
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Eigenvalue of a unilateral shift operator

Let $S:H\to H$ be a unilateral shift operator. I preferred in Example2.3.2 of Murphy's C*-algebras and operator theory that S has no eigenvalues. While $\{\lambda \in \Bbb C ; |\lambda|<1\} \subset ...
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Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
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As $\lambda \to \infty$, $||R_{\lambda}(T)|| \to 0$

"Let $T \in B(X,X)$. Prove that $||R_{\lambda}(T)|| \to 0$ as $\lambda \to \infty$." We have that $R_{\lambda}(T)x=(T-\lambda I)^{-1}(x)$. The problem doesn't specify but the book says that they will ...
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Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
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Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
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Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
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Spectrum of Multiplication Operator $T$ in $C[0,1]$

"Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$." This is an exercise from a PDF of notes I found ...
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Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...
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Proof that the spectrum of an element of a Banach Algebra is non-empty

I don't see why the line indicated with ***** in the following proof is true in the proof that spectrum of an element of a Banach Algebra is non-empty (Arveson, p.27) : For every $\lambda_0 \not\in ...
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Spectrum of Unitary Operators

Let $T_1$ and $T_2$ be two unitary operators. Is it true that the spectrum of $T_1+T_2$ is contained in the closed disc of radius 2?
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Spectral methods with linear programming

Is it possible to model and solve some fundamental spectral methods (say Singular-Value Decomposition) with (Integer?) Linear Programming? Update: say you want to do SVD. Can you model it as a ...
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Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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pertubation of an operator by orthogonal projection

Let $G$ be an operator with compact resolvent on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$ My question ...
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Reference for Spectral theory

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a "gentler" book to learn about these things? Thank you.
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Eigenfunctions of a second derivative operator

Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\alpha >0$ is a positive real ...
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Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
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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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wave propagation - Memory variables and weak formulation

Working on the acoustic wave equation (not in terms of displacement $u$ but in displacement potential $X$ s.t. $\rho u = \nabla X$), I'm trying to get a proper viscoacoustic model by the mean of ...
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Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal ...
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spectrum of compact operators

Let $\phi\in\ell^\infty(\mathbb{N})$. For $p\in[1,\infty]$, define $$M_\phi:\ell^p\to\ell^p,\quad f\mapsto\phi f.$$ Use spectral theory to show that, if $M_\phi$ is compact, then $\phi\in c_0$. Here ...
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Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...