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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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Is it true that $\sigma_{ac}(\bigoplus_{j = 1}^\infty T_j) = \overline {\bigcup_{j = 1}^\infty \sigma_{ac}(T_j)}$?

Let $(H_j)_{j \in \mathbb N}$ be Hilbert spaces and define self-adjoint operators $T_j : \mathcal D(H_j) \to H_j$. Define the operator $T = \bigoplus_{j = 1}^\infty T_j$ on $\mathcal D(T) = \{(\phi_j)...
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Spectral theorem for isometries

I will soon have an exam, and there is something that I simply don't understand: the spectral theorem for orthogonal matrices/endomorphisms (isometries). $\phi$ is an isometry $\Leftrightarrow$ There ...
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Spectral abscissa of a real matrix achieved at a real eigenvector

Consider a real matrix $A\in\mathbb{R}^{N\times N}$ with eigenvalues $\left\{\lambda_i\in\mathbb{C}\right\}_{i\in[N]}$ and the spectral abscissa $\alpha(A)=\max_{\lambda_i(A)}Re(\lambda_i)$ achieved ...
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Every diagonalizable matrix is self adjoint

I have to prove the following: "If $A$ is diagonalizable over $\mathbb{R}$ then there exists an inner product on $\mathbb{R}^n$ such that $A$ is self-adjoint in regards to that inner product.&...
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Simple condition for essential spectrum of self adjoint operator

I am given a bounded self adjoint operator $H:\mathcal{H}\to \mathcal{H}$ and I was wondering whether the following condition implies existence of the essential spectrum. The condition is that there ...
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Is the resolvent of a local operator local?

Let $A$ denote a bounded linear operator on the Hilbert space $l^2(\mathbb{Z})$. We call $A$ a local operator if and only if there exists a $C \geq 0$ such that $\langle e_x | A | e_y \rangle = 0 $ if ...
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Completeness meaning (complete basis vs complete metric space)

Today my professor started talking about the formalism of QM. We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
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Basis of eigenfunctions and spectrum

I have a linear, closed and densely defined operator $A$ defined on a Hilbert space $H$. I prove that the point spectrum consists of discrete eigenvalues, and that the eigenfunctions of $A$ and of $A^*...
Judicaël Mohet's user avatar
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eigenvalues of contraction

Assume $A - B$ is a contraction, i.e., its spectral radius is smaller than $1$. We also assume $A$ is diagonalizable. I am trying to show that if $x^TB=0$, then $x^T$ can be written as a sum of ...
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Spectrum of the laplacian outside of a compact

Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
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Prove that $A$ is a linear operator mapping from $\ell^2$ to $\ell^2$. Determine $\|A\|$, $A^*$ and $\sigma(A)$.

For every $x = (x_n) \in \ell^2$, let $$ Ax = \left(x_1, \frac{x_2}{2}, x_3, \frac{x_4}{2^2}, x_5, \frac{x_6}{2^3}, \ldots \right). $$ Prove that $A$ is a linear operator mapping from $\ell^2$ to $\...
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Decomposition of Spectrum of $U^{-1}TU$, where $U$ is unitary and $T$ selfadjoint

Suppose we have two Hilbert spaces $H_1,H_2$ and a densely defined self-adjoint operator $T:H_2\supseteq D(T)\to H_2$ as well as an unitary operator $U:H_1\to H_2$. Define $S:=U^{-1}TU:U^{-1}D(T)\to ...
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Spectrum of operator in $l^2$ equals closure of $\{b_k : k \in \mathbb{Z}^+\}$

I'm trying to understand why, in the following example, the spectrum of T includes limit points. Let $b_k \rightarrow l$, yet $(T-l \cdot I )(a) = (a_1(b_1-l)\cdots, a_k (b_k-l), \cdots )$. Since $\...
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On self-adjoint extensions and multiplicity of eigenvalues

I hope you can help me with the following question. Let $B$ be a densely defined closed symmetric operator on a infinite-dimensional separable complex Hilbert space $\mathcal{H}$ with deficiency ...
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spectral theory of pseudo-differential operators of class $S^m$

I would be very grateful if you could give me the titles of books that deal with the spectral theory of pseudo-differential operators of class $S^m$ Thank you very much.
Fadil adil's user avatar
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Defining operator as a series

If $e_1, e_2,\ldots$ is a sequence of orthonormal vectors in $H$, $H$ a Hilbert space. Moreover $\mu_1,\mu_2\ldots$ is a sequence of real numbers. If I define an Operator by $Ax:=\sum\limits \mu_i e_i ...
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Quantum Ergodic Theorem: why $\sqrt{-\Delta}$ is used instead of $-\Delta$?

I'm studying the proof of Quantum Ergodic Theorems in the book Partial Differential Equations II: Qualitative Studies of Linear Equations (3rd edition) by Michael E. Taylor. The book includes the ...
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Spectrum of operator $T:l_1\to l_1, Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$

What is the spectrum of an operator $T: l_1 \to l_1$, $x = (x_1,x_2,...,x_n,...)$, $Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$? For $\lambda \ne 0$, equation $Tx=\lambda x$ doesn't have non-zero solutions....
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Prove that $A$ is a bounded automorphism of the vector space $X$. Also prove that $\sigma(A) = \{ \lambda \in \mathbb{C} \mid |\lambda| = 1 \}$.

Let $X = \{ f : \mathbb{R} \to \mathbb{C} \mid f \text{ is a bounded function} \}$. We know that $X$ is a Banach space equipped with the norm $\| f \| = \sup_{x \in \mathbb{R}} | f(x) |$. Let $c \in \...
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A self-adjoint, compact, non-negative, operator is non injective if and only if it is finite rank

Is the following statement true? A self-adjoint non-negative compact operator on a Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle)$ is injective if and only if the closure of its range is ...
DimSum's user avatar
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Dirac Points: Triangular vs. Honeycomb Lattices

I'm reading the paper 'Honeycomb Lattice Potentials and Dirac Points' by Fefferman&Weinstein. To my understanding they claim that the existence of Dirac Cones at K/K' points is entirely determined ...
Julian's user avatar
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Sign of product of two integrals

Let $\Omega$ be an open bounded regular subset of $\mathbb{R}^N$. Let $\lambda_k$ with $0<\lambda_1<\lambda_2\leq\lambda_3\leq...\uparrow\infty$ be the sequence of eigenvalues of $-\Delta$ in $\...
Mathslover's user avatar
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Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$.

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$. (a) Show that from $A^4 = 0$ it follows $A = 0$. (b) Find the eigenvalues of the operator $A$. (c) ...
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Restrictions on a set to be the spectrum of a 1D (discrete) Schrödinger operator.

What restrictions are there on a compact set $E\subset\mathbb{R}$ for $E$ to be the spectrum of a bounded (discrete) Schrödinger operator on $l^2(\mathbb{Z})$? Is there a known necessary and ...
Mathmo's user avatar
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Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??

I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form): $\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$ Can anybody explain to me why ...
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Spectral analysis for harmonic oscillator operator?

Let $L=-\frac{d^2}{dx^2}+x^2, x\in\mathbb R$, the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^2(\mathbb R)$. I am looking for a reference that deals ...
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Neumann series expansion for the resolvent at the spectral radius

Let $T$ be a bounded linear operator on a Banach space. We have following formula for the resolvent: $$\frac{1}{\lambda I-T}=\frac{1}{\lambda }\frac{1}{I-\frac{T}{\lambda }}=\frac{1}{\lambda }\left(I+\...
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Linear operator as linear combination of positive operators [closed]

Lemma: Let $V$ be vector space with inner product. If $S\in L(V)$ is self adjoint, there exist positive linear operators $T, M\in L(V)$ such that $TM=MT=0$ and $S=T-M$. If $V$ is a complex vector ...
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Positive semidefiniteness of Laplacian of undirected graph

Let $D$ be a directed graph. Given a directed edge $e:x\to y$ of $D$, the head of $e$ is defined to be the vertex $y$ and its tail is defined to be the vertex $x$. The gradient matrix of $D$, denoted $...
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Find one quartic root of a matrix

I have found the previous spectral decomposition of the matrix $$A=\begin{pmatrix} 1 & 1 & 0 \\ 0&1&1\\ 1&0&1 \end{pmatrix}.$$ You can see I verified such decomposition indeed ...
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Spectrum of $T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, 4x_{0}, (3x_{-1}), x_{0}, x_{1}, x_{2},\cdots )$

Find spectrum, eigenvalues and eigenvectors of operator $T:\ell^2(\mathbb{Z})\to \ell^2(\mathbb{Z})$, defined by $$T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, ...
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Prove that a necessary and sufficient condition $RQ(X):=\lambda_{max}$, where $\vert\vert x \vert \vert =1 $, is that $Ax=\lambda_{max}x$

I am trying to prove this question my current attempt is as follows. NB $RQ(X):=\langle Ax,x \rangle$, with $A$ an adjacency matrix of a graph $G$ on $n$ and $m$ vertices. ($\Leftarrow$) $Ax=\lambda_{...
andimon's user avatar
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Problem with an example in Werner Amrein book

There is something I don't understand in example 4.13 of Werner Amrein's book "Hilbert Space Methods in Quantum Mechanics". Just before equation (4.55) he says that: if $\lambda\in\text{...
Thomas Belichick's user avatar
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Determine the spectrum of the operator $A$. Is the spectrum composed only of eigenvalues of the operator $A$?

On the space $l^2$, define the linear operator $A$ with the prescription $A(x_1,x_2,x_3,x_4,...)=((x_1+x_2)/2,(x_2+x_3)/2,(x_3+x_4)/2,...)$. (a) Prove that $A$ is a bounded operator on $l^2$ and ...
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Let H be a Hilbert space and $A \in B(H)$. Prove or disprove the following statements.

Let H be a Hilbert space and $A \in B(H)$. Prove or disprove the following statements: (a) If $||A|| = 1$, then $0 \in \sigma(A)$. (b) $\sigma(A^*) = \overline{\sigma(A)}$. Attempt: I actually have no ...
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Prove that $\Sigma_A(\overline A)\leq\operatorname{hd}(A,\overline A)\leq\operatorname{md}(A, \overline A)$.

Let $A\in \mathbb C^{n,n}$ and perturbed matrix $\overline{A}\in \mathbb C^{n,n}.$ Suppose spectrum of A, $\sigma(A)=\{ \lambda_1,\lambda_2,...,\lambda_n\}$ and $\sigma(\overline A)=\{ \overline\...
Unknown x's user avatar
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Prove that $\sigma(\overline A)$ covered by $\cup_{i=1}^n B(\lambda_i, \Sigma_A(\overline A)).$

Let $A\in \mathbb C^{n,n}$ and perturbed matrix $\overline{A}\in \mathbb C^{n,n}.$ Suppose spectrum of A, $\sigma(A)=\{ \lambda_1,\lambda_2,...,\lambda_n\}$ and $\sigma(\overline A)=\{ \overline\...
Unknown x's user avatar
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Spectre map is continuous under hausdorff-metric

could anyone provide me a hint on how to prove the following: Let X be a Banach space and $d_H$ be the Hausdorff metric defined on the compact subsets of $\mathbb{C}$, $$ d_H(M,N):=\max ( \sup_{x \in ...
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Proof that the hausdorff distance $d_H( \sigma (A), \sigma(B)) \le r(A-B)$ for commuting operators $A,B$

I am supposed to prove that for two bounded commuting operators $A, B$ on a banach space, the hausdorff distance of the spectra $d_H(\sigma(A),\sigma(B))$ is less than or equal to the spectral radius ...
Julius Himmel's user avatar
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Product of Laplace-Beltrami eigenfunctions

Is there a simple proof (or counterexample) of the following: for a compact (Riemannian) symmetric space for which $\Delta f = f$ has no Nonzero solution, product of any two nonzero Laplace Beltrami ...
Areon's user avatar
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If $[X,Y]=Y$ must $Y$ have $0$ as the only eigenvalue? [duplicate]

I was asking myself whether, if $Y\in M_n(\mathbb C)$ is such that $XY-YX=Y$ for some other $X\in M_n(\mathbb C)$, then the spectrum $\text{eig}(Y)$ must be $\{0\}$. I think that this is true for $n=1,...
wakewi's user avatar
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spectrum of translation operator

Let $\displaystyle A: c \rightarrow c, (Ax)(n) = \frac{x(n+2)}{n^2}$. The task is to find point spectrum, continuous spectrum, and residual spectrum of $A$ and of adjoint operator $A^*$. $c$ is Banach ...
GeoArt's user avatar
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Proving rank of $A-\lambda vv^t $ is $r-1$.

Let $A$ be an $m\times m$ symmetric real matrix of rank $r$ s.t. $r\ne m$.If $\lambda$ nonzero is an eigenvalue of $A$ with corresponding unit column vector $v$ s.t. $Av=\lambda v$.Then prove the ...
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Characterization of normal Fredholm Operators

I'm working on the following problem, but I'm getting stuck on the last part. Here is the problem: Let $T \in \mathcal{B}(H)$ be normal. Prove $T$ is Fredholm if and only if $0$ isn't a limit point ...
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Spectral measures and spectral families

I've been reading Werner Amrein's book "Hilbert Space methods in quantum mechanics", but I get stuck at the beginning of Section 4.2 When he defines a spectral family of operators $\{E_{\...
Thomas Belichick's user avatar
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Continuous spectrum of an operator $l^1$ to $l^1$

A linear operator $T: ℓ^1 → ℓ^1$ is defined by $$ T(x_1, x_2, x_3, …)= (y_1, y_2, y_3, …), \\ \text { where } y_k=\frac{k+1}{k}x_{k+1}   \text { for } k ⩾ 1 . $$ I want to compute $σ_c(T)$. To ...
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About the spectrum family of the multiplication operator

Let $<,>$ be the inner product of $L^2(\mathbb{R})$. For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
neconoco's user avatar
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Confusion about smoothness and decay of Fourier-like series on a manifold

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
geometricK's user avatar
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2 answers
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Theorem on non-diagonalisable matrix [closed]

My professor gives me the theorem on non- diagonalisable matrices: Let a matrix the $A \in M_{n\times n}(\mathbb{R}).$ $A$ has $k$ independent eigen vectors $\Leftrightarrow$ A is similar to $$ \begin{...
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Does this integral operator between Banach spaces have a non-trivial kernel?

In a question which was recently asked, the goal was to (dis)prove that the set of functions which satisfy a certain equation is trivial. There is already a very neat solution there, but I came up ...
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