Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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What's wrong with this spectrum of a “scalar product” in $l^2$?

Let $T\in B(l^2)$ be s.t. $Tx=(\alpha_1 x_1, \alpha_2 x_2, \cdots )$, where the set of all $\alpha$ is dense in $[0,1]$. I've shown that the set of all eigenvalues is $A=(\alpha_j)_1^\infty$. The ...
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Recovering a Matrix knowing its eigenvectors and eigenvalues

Given the eigenvalues and eigenvectors of a matrix $R^{n\times n}$ is that possible to recover the same matrix from smaller matrices $R^{(n-1) \times (n-1)}$ where one of its eigenvalues and ...
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Spectral measures

Let $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a spectral measure on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Assume also that $E$ is compactly supported in the sense that ...
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Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
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239 views

Decompose a complex symmetric matrix to retain positive definitness

I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that: $$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
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Normalized Cuts and Spectra

I'm looking for a fleshed out proof of the following theorem. Theorem: Let $G=(V,\mathbf{W})$ be an undirected, edge-weighted graph with normalized Laplacian $\mathbf{L}_N$. Furthermore, let ...
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104 views

Finding the spectrum of the Schrodinger operator

Let $H(f) = -f'' + V(x) f$ be the Schrodinger operator on $\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator $H$ in $L^2(\mathbb R)$ for various choices of $V$. In ...
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$(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. ...
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If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
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$\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
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Simple question on self-adjoint operators

Let $H$ be a complex Hilbert space, $T\in H'$ and $T=T^*$. Here is where I need help: If $\sigma(T)\subset\{0,1\}$ then $T=T^2$. Using the spectral theorem I know that $\{0,1\} \supset q(\sigma(P)) = ...
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97 views

Spectrum of a shift composed with a multiplication operator on a vector valued Banach space

Let us consider the space $L_2(\mathbb{R} \times [0,1]; \mathbb{R}^n)$, i.e functions taking values in $\mathbb{R}^n$ and in $L_2$ . Suppose $T$ is a bounded linear operator defined as follows: ...
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840 views

How to understand spectral decomposition geometrically

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = ...
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Eigenvalues of Hilbert-Schmith operator

I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, ...
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102 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
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374 views

Neumann series and spectral radius

I have a question about the convergence of the Neumann series: Let $A$ be a matrix with spectral radius $\rho(A)<1$, i.e., all eigenvalues of $A$ are strictly less than $1$. Does that imply that ...
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189 views

operator norm and spectral radius

is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1? many thanks for any help, it is much appreciated!
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288 views

Spectrum and point spectrum of this operator

Let $T\in \text{Aut}(\ell^2(\mathbb{C}))$ and $T(x)=(a_1 x_1, a_2 x_2,\ldots)$ where $a=(a_i)_i \in \ell^\infty(\mathbb{C})$. How can I easily see what is $\sigma(T)$ and $\sigma_p(T)$ (that are ...
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173 views

Examples for spectrum of an operator

Looking for easy-to-understand examples for the spectrum of an operator, preferably so that they exposed some special properties. The right shift is a nice example of an operator which does not have a ...
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Fourier transform physical meaning [closed]

What is the physical meaning of the Fourier transform expressed at the spectral density? Also, what is the relationship between the Fourier transform and the total energy of an oscillating system? ...
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90 views

Difference in sound between a string and a pipe

I am told that I can model the vibration of a guitar string of length $l >0$ by the following Sturm-Liouville equation $$ -u'' = \lambda u \: \: \text{ on } [0,l],$$ with boundary conditions ...
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116 views

closed operator, projection

Let $A: D \subset X \to X$ be a closed linear operator. X is a Banach space. Furthermore we have $\gamma: [0,1] \to \mathbb{C}$, $\gamma$ is a $C^1$ curve and $\gamma \subset \rho(A)$, where $\rho(A)$ ...
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Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
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Compute Rayleigh quotient for ODE

I am trying to find Rayleigh quotient for this equation: $u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where $0 \le r \le 1$. Is there any way to compute eigenvalue ...
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what is the largest eigenvalue of the average of non-negative matrices?

I have a set of square matrices $A_i \in \mathbb{R}^{n \times n}$ for $i=1,\ldots,N$, such that $[A_i]_{jk} \ge 0$ for all $i$ and coordinates $j,k$. If the largest eigenvalue of each $A_i$ is ...
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Tensoring Spectral triples that are composed from Real algebras.

I have a misunderstanding that I am hoping is really quite trivial. In connes standard Non-commutative geometry model of electroweak interactions he takes the algebra input in his finite spectral ...
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Eigenvalues of the “Laplacian” on [0,2$\pi]\subset\mathbb{R}$

for the dirichlet eigenvalue Problem on compact and connected Riemannian manifolds, the eigenvalues of the laplacian consists of a discrete sequence. On the other hand, if we consider ...
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Infimum of the spectrum of an unbounded selfadjoint operator

Let $A$ be an unbounded selfadjouint operator in the Hilbert space $H$, having domain $D(A)$. Denoting by $\sigma_A$ the spectrum of $A$, we have $\inf \sigma_A \ = \ \inf_{u\in D(A),\|u\|=1} \ ...
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Fixed-point method in many-dimensions

A well known method of easily solving multi-dimensional non-linear root finding problems, is to bring the equations into the form: $$\bf x = g(x)$$ And then iterating. The problem is, one has to find ...
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Uniform Poincaré-Wirtinger constant for diffeomorphic domains?

Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
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can I decide the following from the inverse of I-A?

I have a square matrix $A$. Is it possible to determine if its largest eigenvalue is smaller (by magnitude) than 1 by inspecting the matrix $(I-A)^{-1}$? (we can assume that $I-A$ is invertible.) ...
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237 views

Spectrum of this Operator

Let $A\colon \ell^{1}\to \ell^{1}$ be defined by $A(x)=(x_{2}+x_{3}+x_{4}+ \dots,x_1,x_2,x_3,\dots)$ where $x\in\ell^1$ iff $\sum|x_k|<\infty$. Let $D$ be the closed unit disc in $\Bbb C$ and ...
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How can I calculate the eigenvalue of the following matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ such that its elements are all non-negative values. I know that for any $k$, $A^k$ has elements on the diagonal which are smaller or equal to 1. Can I ...
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218 views

is there a way to find or upper bound the largest eigenvalue of the following matrix?

I have a matrix $A \in \{0,1\}^{n \times n}$ -- i.e. a matrix with 1s and 0s only. Is there a way to find or upper bound its largest eigenvalue? I have a feeling it is related to connectivity of ...
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Finding Spectral Radius of Matrix

Find the Spectral Radius of $A=$ $\mbox{} \left[ \begin{array}{cc} 1 & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & -c & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} ...
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343 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
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266 views

spectral radius.

I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem? The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ...
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64 views

Proof that these Hessian matrix identities are similar matrices

I am wondering if $Q, P$ are similar matrices where for a function $f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$ $Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$. ...
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211 views

Similar matrix proof

$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
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506 views

The spectrum of an unbounded operator

It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their ...
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spectrum of convex combination

$A,B$ are $n\times n$ Hermitian matrices. If the eigenvalues of $A$ and $B$ are all in an interval $I$, then the eigenvalues of any convex combination of $A,B$ are also in $I$. In the book Bhatia, ...
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Strict positivity on the diagonal of a particular integral kernel: A question from Simon's Schrödinger Semigroups

This is a question pertaining to a (formerly?) open question from Barry Simon's Schrödinger Semigroups. In Theorem C.5.2 (page 504) of that publication, the existence of a specific function ...
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is it true that the infinity norm can be bounded using the $L_2$ norm the following way?

Let $v \in \mathbb{R}^k$, and let $A \in \mathbb{R}^{m \times k}$ and let $B \in \mathbb{R}^{m \times n}$ such that each column of $B$, $B_i$, has $$||B_i||_2 \le 1.$$ Is it true that: $||v ...
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Reading a DFT plot - did I get this right?

I am simulating the evolution of a liquid film through the solution of a 4th order nonlinear partial differential equations. Of late, I began experimenting with DFT of the result that I have. My ...
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Root Convergence rate of Iterative Scheme [closed]

I have an iterative sequence for optimizing an EM algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and ...
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219 views

Spectrum and orthogonal projection

Let $G$ be an operator on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let also $P$ be the orthogonal projection onto $\ker G$, $G_1$ the restriction of $G$ on $\ker P$ and $G_2$ ...
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210 views

Inequality for a selfadjoint operator on Hilbert space

Let $T$ be a (possibly unbounded) selfadjoint nonnegative operator on a Hilbert space $H$ with domain $D$. Assume that $\langle T u, u \rangle \leq c$ for some $c>0$ and some $u\in D$. I found ...
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When will the equality of Neumann's Trace inequality holds?

I am now studying the Neumann's trace inequality. Some of the literature said that the equality holds when the two matrices have simultaneous ordered spectral decomposition. Yet they don't really ...
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Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
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Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...