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

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Convergence of $X_{n+1}=A X_n+C$.

somebody can help me to understand the convergence of this equation, $X_{n+1}=A X_n+C$ where $A$ is an $n \times n$ matrix, $C$ is a vector and spectral radius is $\leq 1$. I like to understand ...
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In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
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1answer
16 views

Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: ...
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26 views

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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24 views

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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1answer
18 views

Are they true these generalizations from matrices to operators about functional calculus?

Motivation: If we have some real function $f$ defined on an interval $I$ and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ is a diagonal matrix such that $\lambda_i \in I$ for all $1\leqslant i ...
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451 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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How can I tell that my matrix is nilpotent?

I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ...
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28 views

spectral measure and normal operators range

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$. Is this true? Let ...
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Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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normal operators spectral decomposition

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$.
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+400

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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1answer
19 views

what is a spectral function?

My knowledge in spectral theory is very limited, but lately I heard talking about the spectral function of an operator and how it's important. By curiosity I tried to look for a definition and a ...
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1answer
42 views

Proof that a random measure with orthogonal increments is a measure

Let me first state what I mean by a random measure with orthogonal increments. Definition: A random measure with orthogonal increments $Z$ is a collection $\left(Z(B): B \in ...
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43 views

spectral measure of normal operator's eigenvalue [closed]

Let $H$ be a Hilbert space and and $T \in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e., the set of all eigenvalues of $T$) and let $E$ denote the spectral measure. Show that if ...
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2answers
38 views

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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13 views

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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How can we prove that the spectral radius $\rho(A)^k$ is equal to $\rho(A^k)$.

How can you prove for a square matrix $A$ that $\rho(A)^k = \rho(A^k)$ with $\rho(A)$ = the spectral radius of $A$? Thanks!
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1answer
30 views

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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1answer
26 views

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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1answer
17 views

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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1answer
63 views

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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Need mathematical explanation for different musical notes sound different on different instruments [migrated]

I am not expert in music. There are number of musical instruments. One (especially a person who knows about music) can blindly recognize which instrument is being played just by listening to it. I ...
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1answer
29 views

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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Does spectral norm of a square matrix equal to its largest eigenvalue in absolute value?

I have one simple question. Given the spectral norm $\left \| . \right \| _2$ of a matrix $A$, which is equal to the squareroot of the largest eigenvalue of $A^{^*}A$ $$\left \| A \right \| ...
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Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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51 views

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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52 views

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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58 views

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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Domain of closed unbounded operator

Let $A$, $B$ be two closed unbounded operators such that: (1) there exists dense subspace $\mathcal{D}$ of $Dom(B)$ which is contained in $Dom(A)$, (2) for every $\psi \in\mathcal{D}$ it holds $$ ...
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First representation theorem for sesquilinear forms - what is the role of the “core”?

In the first representation theorem, the notion of the core of a sesquilinear form appears. What is the intuition behind this notion, in context of this theorem and in general? I appreciate any ...
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If a normal element of a C* algebra has real spectrum, then it is self-adjoint

Let $A$ be a $C^*\!$-algebra. Suppose $x$ is a normal element of $A$ and $\operatorname{spect}(x)$ lies in $\mathbb{R}$. Prove that $x$ is self-adjoint. I tried the following: using ...
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Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
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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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Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
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Solving for the spectrum and eigenvectors of the “shift operator(?)” $T$ in $P_3(\mathbb{R})$?

This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$ (The actual question can be ...
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Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
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Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
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Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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What do we know about inverses of matrices which are “like” Laplacians of graphs?

Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...
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Spectrum of double infinite shift using isometry to Fourier series

I'm trying to find the spectrum of the operator $T: l^2(\mathbb{Z}) \to l^2(\mathbb{Z})$ given by right shift but I am having some difficulties. I can show that $l^2$ is isomorphic to ...