# Tagged Questions

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

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### Spectrum of weighted shift operator

The Banach space considered is the following: $(l^{\infty}(\mathbb{Z}), \|\cdot\|_{*})$ with $\|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|$. Define $A$, an ...
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### Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of ...
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### How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
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### What do we know about rank-2 perturbations?

Are there any theorems known about the changes in spectrum of a matrix A when it is changed to A+X, when X is rank-2? I am particularly interested in the case when X is a zero matrix except for ...
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### Subspace Perturbation

For two positive semidefinite matrices $A,B\in\mathbb{R}^{n\times n}$, with dominant $r$ dimensional subspaces $U,V\in\mathbb{R}^{n\times r}$ and eigenvalues $\Sigma_A, \Sigma_B$, what can we say ...
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### Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...
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### cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
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### Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
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### Self-adjoint operator has non-empty spectrum.

I am trying to prove, that a self-adjoint (maybe unbounded) operator has a non-empty spectrum. So far I have argued, that if $\sigma(T)$ would be empty, $T^{-1}$ would be a bounded self-adjoint ...
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### Spectral theorem for compact and self-adjoint operators

I am looking at the proof of this theorem which states that if $H$ is a separable Hilbert space and $A:H\rightarrow H$ a compact self-adjoint operator, then there exists a sequence of real eigenvalues ...
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### What is the spectrum of the sequence operator $B: (x_1,x_2,\ldots) \rightarrow (0,x_1,\frac{1}{2}x_2,\ldots,\frac{1}{n}x_n,\ldots)$?

The question is stated in the title, and the operator is defined on $\ell^2$. I have determined that $||B|| = 1$, and therefore $\sigma(B) \subset \{\lambda \in \mathbb{C} : \, |\lambda| \le 1 \}$. ...
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### How to extract power of top two frequencies of a spectrum without using an FFT?

What I'm trying to do is see if a particular frequency component becomes dominant (and I don't really know what the dominant frequency is). Therefore, I figured that I can get the top two peaks of ...
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### Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
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### On important functions relflecting spectral properties of Jacobi operators

The spectral analysis of Jacobi (semi-infinite, tridiagonal) operators acting on $\ell^{2}(\mathbb{N})$ is deeply investigated. A crucial role is played by function $m$ which is usually known as Weyl ...
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### Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
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### Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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### Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
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### Is the structure constant additive on connected components?

Definition of the Structure Constant Let $M$ be a Riemann surface and $\mu$ a smooth metric on it; let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and ...
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### Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
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### Difference between continuous and essential spectrum, examples?

Can anyone help me to understand the definitions of the continuous and essential spectra in simple terms and point out the difference on examples where the two definitions coincide and where don't? ...
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### Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
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### How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to ...
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### Method for ?not quite? weighted least squares fitting for more realistic results

I need a linear least squares type of fitting algorithm that understands how to weight the probability of a response coming from certain functions over another. To explain, given the standard linear ...
In my course of spectral theory and operator algebras I came across the following exercise: Let $\mathcal{A}=C_0(X)$ where $X$ is a locally compact Hausdorff space. Describe the multiplier algebra ...
Let $A$ be a real symmetric matrix. Now fix a diagonal index say "i" and let $x > max-eigenvalue(A)$. Now is there any thing known about the Gershgorin circle of $[1/(x-A)]_{ii}$ in terms of the ...