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

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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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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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spectrum of weighted translation semigroup

My Banach space is $\mathcal X=\rm{L}^1(\mathbb R_+)$. I would like to know the spectrum of $A\phi(x)=-\phi'(x)-f(x)\phi(x)$ on $D(A) = \{g\in\mathcal X,\ g\text{ absolutely continuous}, g(0)=0\text{ ...
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How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
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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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Convergence of $X_{n+1}=A X_n+C$. [on hold]

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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20 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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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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70 views

Spectral Measures: Embedding

This thread is just a note! Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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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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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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Is there a residue theorem for holomorphic operator-valued functions?

I'm wondering whether there is such a thing as a "residue theorem for holomorphic operator-valued functions". More precisely, I want to evaluate an integral of the form $P:=\int_{\Gamma} (A(\lambda) ...
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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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1answer
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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2answers
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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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Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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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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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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1answer
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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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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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Normal Operators: 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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2answers
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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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How is the study of fractals related to Fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies), but to my dismay, ...
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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
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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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1answer
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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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1answer
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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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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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Normal Operator: Empty Spectrum

Given a Hilbert space $\mathcal{H}$. For normal operators: $$N^*N=NN^*:\quad\sigma(N)\neq\varnothing$$ How can I check this?
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1answer
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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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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
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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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Summary: Spectrum vs. Numerical Range

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Denote for shorthand: ...
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1answer
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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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Spectral Measures: Uniqueness

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E^{(\prime)}:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote their operators by: ...
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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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Spectral Measures: References

I am trying to learn a little bit about the spectral theory of unbounded operators but the textbook we are using (Birman and Solomyak: Spectral theory of Self-Adjoint Operators in a Hilbert Space) is ...
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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 ...
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1answer
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Spectral Measures: Stone's Formula

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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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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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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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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1answer
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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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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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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 ...