Questions tagged [spectral-theory]

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

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Singular values of a random matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
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18 views

$Ran(A-\lambda)^{\perp} = Ker(A-\lambda)$ for self-adjoint operator $A$?

I am reading through Theorem 5.5 in Hislop and Sigal, and I have the following confusion about the proof for item 2. My confusion is, why can you immediately conclude that \begin{equation}\label{key} ...
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Confusion regarding spectral decomposition of normal operator

I'm reading out of Operator theoretic aspects of ergodic theory, and specifically looking at pages 380 and 381, and specifically at Remark 18.14. The authors have already proven the existence of a ...
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If $Re \lambda \geq 0$ for all $\lambda \in \sigma(A)$ show $Re\langle x,Ax\rangle\geq 0$

Let $H$ be a Hilbert space and A a normal operator with the property $Re\lambda\geq 0$ for all $\lambda \in \sigma(A)$. We want to show that $Re\left<x,Ax\right> \geq 0$ for every $x \in h$. I ...
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Does the spectrum of a bounded from below self-adjoint operator have a lower bound?

Let $A$ be a self-adjoint operator in a complex separable Hilbert space that is unbounded, but bounded from below, i.e. $$\exists m>-\infty, ~\forall f\in D(A) \subsetneq \mathcal H , \langle Af, f ...
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45 views

Generalization of the Spectral Theorem to arbitrary matrices

Let $N\in M(n, \mathbb{C})$ be an $n \times n$ normal matrix, meaning that $N^\ast N = NN^\ast$. The spectral theorem states that $N$ is unitarily diagonalizable $$ N = U^{-1} \Lambda U,$$ where $U$ ...
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26 views

Spectral radius of a matrix of bounded operators

If $a,b,c,d$ denote four bounded operators on Banach spaces, let $$ A=\begin{pmatrix} a&b\\ c&d \end{pmatrix} $$ Can the spectral radius of $A$ be linked to those of $a,b,c,d$ ? For example, ...
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Does PT-symmetry breaking always occur at an exceptional point?

For a parametrized family of matrices $M(k)$ an exceptional point is defined to be a critical point $k=k_c$ where the number of eigenvectors of the matrix is less than the dimension of the matrix. ...
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Symmetry breaking in pseudo Hermitian matrices

A matrix $M$ is pseudo-Hermitian if it satisfies $$M^\dagger = \eta M \eta^{-1},$$ where $\eta$ is a Hermitian invertible matrix. The spectrum of pseudo-Hermitian matrices is either completely real or ...
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Iff conditions that f(a)=g(a) for a nilpotent element $a$ in a Banach algebra

Let $\mathcal{A}$ be a Banach algebra with $1$, let $n \in \mathbb{N}$, and let $a\in \mathcal{A}$ such that $a^n=0$. (a) Determine $\sigma(a)$. (b) Let $f,g \in \mathrm{Hol}(a)$. Give a necessary ...
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Spectral theory of self-adjoint operators [closed]

Let H be a Hilbert space, and $ T \in \mathcal {L} (H) $ an self-adjoint operator. So there is a measurement space $ (X, \nu) $ a limited function $ g: X \to \mathbb{R} $ and an unit operator $$ U: L ^...
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Prove the following inequality on a Banach Algebra $\mathcal{A}$

Let $x \in \mathcal{A}$ and $K \subset \mathbb{C}$. Let $r$ be the minimum distance of $K$ from $σ(x)$ (where $σ(x)$ is the spectrum of $x$) and also $r > \delta$ for some $\delta > 0$. Show ...
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Upper bound on spectral radius of sum of two Metzler matrices

Given a Metzler matrix $A$ (non-diagonal elements are positive), I am trying to find an upper bound on the spectral radius of $A+A^T$ (preferably in terms of the spectral radius of $A$). In particular,...
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Why can a self-adjoint compact operator on a Hilbert space be approximated by the linear combination of rank 1 operators?

Hilbert-Schmidt Theorem says that suppose $A$ is a self-adjoint compact operator on the Hilbert space $X$, then $X$ has an orthonormal basis $\{e_i \,|\, i \in I\}$ ($I$ is the index set) which is ...
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How to determine the spectral measurement of a matrix

Let $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ a linear operator whose matrix in the canonical base is $ \begin{pmatrix} 1 & 2 \\ 2 & -3 \end{pmatrix}.$ Find the expression of the spectral ...
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Intuition question about the spectral definition of expander graphs

Disclaimer: I don't know too much about graph theory, and next to nothing about spectral theory. I know this will probably make answering this question difficult, so I am sorry in advance! Hi! I am ...
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Why a closed unbounded operator with compact resolvent has its spectrum consisting of eigenvalues with finite multiplicity?

Could someone tell me (or has a reference) why a closed unbounded operator with compact resolvent has its spectrum consisting of a sequence of complex eigenvalues, each with finite multiplicity?
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Extracting covariance matrix in a Gaussian HMM using method of moments

I'm having a problem extracting an unknown matrix from a part of a larger matrix. The unknown matrix (matrices, really), are state-dependent covariance matrices in a Gaussian HMM. I believe the ...
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Equivalence with Baker transformation

On page 240 of "Methods of Modern Mathematical Physics I: Functional Analysis", Reed & Simon define the Baker's transformation as $$T \langle x,y \rangle = \begin{cases} \langle 2x,...
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Iff conditions for $f(a)=g(a)$ for a nilpotent element $a$ in a Banach algebra

Let $\mathcal{A}$ be a Banach algebra with $1$, let $n \in \mathbb{N}$, and let $a\in \mathcal{A}$ such that $a^n=0$. (a) Determine $\sigma(a)$. (b) Let $f,g \in \mathrm{Hol}(a)$. Give a necessary ...
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53 views

Complex interpolation of spaces defined by operator with bounded imaginary powers

In Corollary 4.6 of this paper https://core.ac.uk/download/pdf/81991766.pdf they seem use a result that if $\Delta : D \to H$ is essentially self adjoint, such that $D \subseteq H$ is a dense subset ...
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37 views

Dimension of eigenspaces of compact, self adjoint operator are finite?

Suppose $H$ is a hilbert space. Let $T:H\rightarrow H$ be a compact (in the bolzano weistrass sense), self adjoint operator. Then show that if $\lambda$ is an eigenvalue then then eigenspace of $\...
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25 views

Inverse operator being bounded and linear

So, here is my next edited attempt to solve this problem. conditions for the existence of the inverse of spectral operator as a bounded linear operator Let $X$ be a Hilbert space with a countable ...
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Relation between spectrums of algebra and subalgebra

I want to ask this. Suppose that $\mathcal{A}$ is a Banach algebra and $\mathcal{B}$ a subalgebra of $\mathcal{A}$. I want to show that if $σ_{\mathcal{A}}(x)$ does not seperate $\mathbb{C}$ then $σ_{\...
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$A$ is a compact and Hermitian operator— if all eigenvalues of $A$ are non-negative, then $\langle Ax,x \rangle \geq 0 $

So, $A$ is defined on a Hilbert space $X$ and all of its eigenvalues are non-negative. Show that $\langle Ax,x \rangle \geq 0, \forall x \in X $. My attempt so far: Since $A$ is compact and Hermitian, ...
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20 views

conditions for the existence of the inverse of spectral operator as a bounded linear operator

Let $X$ be a Hilbert space having a countable orthonormal base $[e_1, e_2, \cdots]$. Also, suppose $Ax = \sum _{n=1} ^{\infty} \alpha_n \langle x,e_n \rangle e_n $, where $[\alpha_n]$ is a real ...
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37 views

How can i find an estimation of the inverse operator $\|(I - T)^{-1}\|$

$T$ is a bounded linear operator such that $\|T\| \leq e^{a}$; where $a > 0$.\ if the operator $I - T$ is invertible,\ How can i find an estimation of $\|(I - T)^{-1}\|$. thanks;
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Eigenvalues of an operator in an infinite-dimensional Hilbert space

This is from Cheney's Analysis for Applied Mathematics: Let $X$ be a Hilbert space having a countable orthonormal base $\{u_1, u_2, \dots\}$. Define an operator $A$ by the equation $$Ax = \sum_{n=1}^\...
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Meaning of does not separate zero from infinity

In this excerpt, it states that ‘ Let $T$ be a bounded linear operator on a Banach space $X$. Suppose spectrum that the $\sigma(T)$ does not separate zero from infinity ( consequently the operator is ...
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Bounded self-adjoint operator satisfying $A^2-A\leq 0$ satisfies $0\leq A\leq 1$?

Let $A$ be a bounded self-adjoint operator on some Hilbert space $\mathcal H$. Assume $A^2\leq A$, meaning that $\langle A^2v,v\rangle\leq\langle Av,v\rangle$ for all $v\in\mathcal H$. I am wondering ...
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Karhunen-Loeve expansion of non-centered processes

The typical form of Karhunen-Loeve expansion is on a detrended stochastic process. E.g., let $Y(t)$ be a stochastic process on $[0,T]$, and let $X(t) = Y(t)-\mathbb{E}Y(t)$ with a continuous ...
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Proving that an expression is zero for all continuous complex valued functions on $[0,1]$

I would like to prove the following identity: $$ \int_{0}^{t} \left [\frac{(t - s)^n}{n!}f(s) - \int_{0}^{s} \frac{(s - x)^{n - 1}}{(n - 1)!} f(x) dx \right] ds = 0. $$ Yes, I know it's really ugly. ...
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51 views

Weyl's law for eigenvalues

I am calculating the eigenvalues of Laplacian on sphere. The Laplace's problem(with the Dirichlet's boundary condition) on sphere is as follows: $$L u + \lambda u = 0$$ where $L$ has the following ...
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eigenvalues and fundamental group

Let us start with a "nice" domain $\Omega\subset\mathbb R^2$ (bounded smooth boundary), and let $\Omega_\epsilon=\Omega$ with a small hole removed. Suppose $K$ is a continuous function on $\...
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74 views

Spectral decomposition of Rodrigues' rotation formula

I am supposed to rewrite Rodrigues' rotation formula $R(v)=v\cos \phi+k(k\cdot v)(1-\cos\phi)+(k\times v)\sin\phi$ in the form of spectral decomposition. I can figure out that the eigenvalues are 1, $...
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Multiplication of a Riesz basis is always a Riesz basis?

Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1)^2$. My question is the following: If we multiply the basis by the matrix $e^{Mx}$ where $M$ is a constant matrix of dimension $2$...
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Spectrum of bilateral shift

Let $T:l^2(\mathbb{Z})\longrightarrow l^2(\mathbb{Z})$ and define $T(\{x_n\})=\{x_{n-1}\}.$ Some reference tells me that the spectrum of $T$ is $\mathbb{T}=\{\lambda\in\mathbb{C}:|\lambda|=1 \}.$ What ...
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How do I show that null sets of a spectral measure are null sets on the induced complex measure?

I'm going through my lecture notes again for functional analysis, and I came up on this property that I can't seem to prove. So $E(\omega)$ is your garden-variety spectral measure, and we define ...
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Writing the Resolvent as an Integral

Let $A$ be a self-adjoint operator on a Hilbert space, and let $z \in \mathbb{C}$ not in the spectrum of $A$, i.e. the resolvent operator $(A - z)^{-1}$ is well-defined (bounded, etc). I want to ...
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Representation of self-adjoint operators as multiplication operators - the spectral theorem.

Let $A$ be a densely defined self-adjoint linear operator on a separable complex Hilbert space $H$. Then, it is often stated (e.g here and theorem 4.1) that $A$ is unitarily equivalent to a ...
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What will be $\sigma (T)\ $?

Let $T$ be an operator on $\ell^2$ given by $Te_n = d_n e_n,$ where $$ d_n = \left\{ \begin{array}{lr} \frac {1} {2}, & \text {if}\ n \equiv 0\ (\text {mod}\ 3) \\ \frac {n} {n+1}, &...
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Why $U$ is unitary in the Lax pair?

I was reading the page of Lax pair on wikipedia, and I cannot see why if $P(t)$ is skew-adjoint, then $U(t,s)$ will be unitary? (it's written in the section of "Isospectral property"). ...
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47 views

Showing that $\|(T\pm i)x\|^2\ge\|x\|^2$ gives $\sigma(T)\subseteq[-\|T\|,\|T\|]$

Let $\mathcal{H}$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. An operator $T\in\mathcal{B}(\mathcal{H})$ is called self adjoint, if $\langle Tx,y\rangle=\langle x, Ty\...
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58 views

Is a spectral projection strongly/norm continuous?

Given a bounded self-adjoint operator $H$ and a number $\mu$, consider the spectral projection $P_\mu$ onto the set $\{x|x\leq\mu\}$. I want to check if the map $\mu\mapsto P_\mu$ is strongly/norm ...
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38 views

Spectral theory for compact normal operators.

Let $T$ be a compact normal operator on an infinite dimensional Hilbert space $\mathcal H.$ Then I came across the fact that the spectrum $\sigma (T)$ of $T$ is either finite or countably infinite ...
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39 views

How is a real symmetric matrix a limit of symmetric matrices with distinct eigenvalues?

In Linear Algebra, Gilbert Strang, $4$th edition the theorem $5$S (section $5.6$) is the following - Every real symmetric A can be diagonalized by an orthogonal matrix Q. Every Hermitian matrix can ...
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How to prove energy of a path graph increases with its increasing path weight?

Given a graph $G$ with $n$ vertices, if its adjacency matrix $A$ has eigenvalues $\lambda_1 \geq \lambda_2 \geq . . . \geq \lambda_n$ then the energy is defined as: $$\mathcal{E}(G) =\sum_{i=1}^{n} |\...
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Spectral Theorem for compact normal operators on Hilbert space from Banach space considerations

Let T be a compact operator on an infinite-dimensional Banach space $E$. Let $\sigma(T)$ denote its spectrum. Then $0 \in \sigma(T)$ and every nonzero point $\lambda$ of $\sigma(T)$ is an isolated ...
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How special is the Cantor-set spectrum of the almost mathieu-operator?

The Almost Mathieu Operator (parametrised by two important parameters, $\lambda,\alpha$) has a spectrum that is apparently the Cantor set when $\alpha$ is irratonal and $\lambda \neq 0$. How ...
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For a bounded operator $A$, show that $(fg)(A)=f(A)g(A)$

Let $f$ be an analytic function in the neighbourhood of the spectrum $\sigma(A)$ of a bounded operator $A$. Let $C$ be a contour, counterclockwise around $\sigma(A)$ within the domain of $f$. Finally ...

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