# 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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### $Ran(A-\lambda)^{\perp} = Ker(A-\lambda)$ 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
### 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 ...