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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Question about proof that normal operators have invariant subspace
In Normal $T\in B(H)$ has a nontrivial invariant subspace, Haskell Curry said (for case (ii)) to pick two open, disjoint subsets of $\sigma(T)$ where $|\sigma(T)|\geq 2$. I don't think that this is ...
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Nonlinear ODE with (pseudo-)spectral method
Consider an ODE of the form
$$c_0 u + c_1 u' + c_2 u'' \enspace = \enspace f(x,u) \quad .$$
I want to solve this ODE with spectral methods. To this end, I approximate $u$ by a series of the form
$$ u(...
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Any concise proof for QR algorithm?
I know many books would have proofs for the QR algorithm, but I wonder if there is a concise one. (e.g. 2-3 pages in PDF)
Here is a statement. Let's assume that any real $n$ by $n$ matrix $A$ can be ...
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Fourier Transform Method for Solving Fredholm Translation-Invariant Covariance-Kernel Integral Equations
I'm examining a problem involving the Fredholm integral equations of the second kind and trying to apply Fourier transform techniques. Particularly, the challenge arises when seeking to express the ...
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How to prove $G_{A^*}\supseteq (V G_A)^\perp$?
If $A:D\subset H \rightarrow H$ on a Hilbert space $H$ is a unbounded densely defined operator. Then its adjoint exists and
$$\left\langle \begin{pmatrix}x \\ Ax \end{pmatrix},\begin{pmatrix}0 & -...
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Prove that the limit of an operator sequence is infinite [closed]
Let $X$ be a Banach space, $G$:={$A$ is an operator : $A$ and $A^{-1}$ $\in$ $L(X)$}.Suppose $T$ is a boundary point of $G$, and ${T_n}$ is an operator sequence which is in $G$ and $L(X)$.$T_n$ tends ...
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essential range of function,range and spectrum
hi here $\sigma$ is spectrum,if $(X,\mu)$ is topological measure owed with borel measure and $$
r_{ess}(f) = \{w \in \mathbb{R}_+ : \mu(f^{-1}(B(w, \epsilon))) > 0\}.
$$
i have proved that if f is ...
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bringing $-i \frac{d}{dx}$ to the form $U^* M_f U$
The operator $-i \frac{d}{dx} : H^1(\mathbb{R}) \subset L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ is self adjoint hence it is unitary equivalent to a multiplication operator
i.e. $-i \frac{d}{dx}= U^...
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The theory of functions of a normal transformation on a real inner product space
This is exercise §82.2 of Halmos 1958, 2nd ed.:
Discuss the theory of functions of a normal transformation on a [finite-dimensional] real inner product space.
In the preceding chapter, general ...
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Representation of function in $W^{1,2}([-1,1])$ by function in $W^{1,2}_0([-1,1])$
I read some notes from a lecture, where it is claimed that
any function $\varphi \in W^{1,2}([-1,1])$ can be represented by
$$\varphi(x)= \psi(x) +c_1e^{x}+c_2 e^{-x}$$
where $c_1,c_2 \in \mathbb{C}$ ...
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Deficiency indices and dimension
Let $H$ be a Hilbert space and $A:D(A)\subset H \rightarrow H$ a linear operator. In what sense are the deficiency indices
$$\mathrm{dim}(\mathrm{ran}(A+ z) )\text{ }(z\in \mathbb{C})$$
defined?
I ...
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John von Neumann theorem on self adjoint extentions
Let $H$ be a Hilbert space and $A:D(A)\subset H \rightarrow H$ be symmetric and closed.
Assume $A$ has a selfadjoint extention $B$. Then the Cayley transform of $A$ has also a unitary extention i.e. ...
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Correspondence between unitary operator and self adjoint extention
Let $A: D(A) \subseteq H \rightarrow H$ be a symmetric and closed operator on a Hilbert space $H$.
Denote by $\mathscr{K}_{\pm}:=\mathrm{ran}(A\pm i)^\perp$ and $n_\pm:=\mathrm{dim}(\mathscr{K}_\pm)$ ...
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Bijection between two sets of Operators
Im reading a script where there is stated that there is a bijection from the set of all
operators $A: D(A)\subset H\rightarrow H$ ($A=\overline{A}\subset A^*$) to the set of all
operators $V: D(V)\...
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Self adjoint operator $A$ satisfies $\|A(A-i)^{-1}\|\leq 1$?
Let $H$ be a seperable Hilbert space and $A=A^*:D\subset H \rightarrow H$ a possibly unbounded linear densly defined selfadjoint operator.
I try to understand a proof in a script, where a crusial step ...
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Compact resolvent implies $\mu_{\infty}(A) = \infty$
Let $A: D(A) \to \mathscr{H}$ be a densely defined self-adjoint operator on a Hilbert space $\mathscr{H}$ which is bounded from below. For each $n \in \mathbb{N}$, define:
$$\mu_{n}(A) := \inf_{\...
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Necessary and sufficient conditions for a transition matrix to have a limit
Let $T$ be an $n \times n$ transition matrix, i.e. the rows sum to 1 and the entries all lie in the interval $[0,1]$. What are necessary and sufficient conditions for which the limit $lim_{n \...
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If $A,B\in GL_n(\mathbb{C})$ are positive matrices. Prove that the following holds
If $A,B\in GL_n(\mathbb{C})$ are positive matrices. Prove that, there is $\lambda\in\sigma(A^{-1}B)$ such that the following holds $$\langle Ax,x\rangle\ge\lambda^{-1}\langle Bx,x\rangle\ \forall x\in\...
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Theorem (Rellich) - Perturbation Theory
I got stuck with part of a proof of: The steps are all clear to me, until it is said:
"Therefore $f_w$ is the eigenvector of $ A+wB$ associated with the eigenvalue lying within for all small ...
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$\mu_{x}(A):=\langle P_Ax,x\rangle$ is also called Spectral measure?
As I know the spectral measure of a linear possibly unbounded self adjoint operator $T$ on a seperable Hilbert space is a projection valued measure defined by its measurable functional calculus $A\...
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unit decomposition
Find unit decomposition of multiplication operator $(Ax)(t) = \sin(t)\,x(t)$ in $L^2(\mathbb{R})$ space.
I started from spectrum of operator $\sigma(A)=[-1,1] \implies
E_\lambda \lambda=E(-\infty,\...
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What properties of an operator can be deduced by looking at its approximate point spectrum?
May I ask something to the community:
If $A: H \rightarrow H$ is a normal, compact operator on a complex Hilbert space H,
then $\sigma(A)=\sigma_p(A)\cup \lbrace 0\rbrace$, since if $z-A$ $(z\neq 0)$ ...
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Suficient condition for an unital $C^*$-algebra homomorphism to be surjective on postive elements.
I am trying to find sufficient conditions for an unital $C^*$-algebra homomorphism $\alpha:A \rightarrow B$ to be surjective on positive elements, that is, $\alpha(A_+)=\alpha(B_+)$. For example, if ...
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Nilpotency of $(A-z)P_z$ imply $z$ is an Eigenvalue?
Let $z$ be an isolated point in the spectrum of some possibly unbounded operator $A:D\subset X \rightarrow X$ on a Banach space $X$. Where the spectrum $\sigma(A)$ without $z$ is closed. Let $P_z$ be ...
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Convergence of self-adjoint operators with converging spectra
Let $A=\int_{\sigma(A)} \lambda dP(\lambda)$ be an unbounded self-adjoint operator on a Hilbert space $H$ with dense domain $D\subset H$.
Let $A_n=\int_{\sigma(A)\cap[-n,n]} \lambda dP(\lambda)$ (see ...
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How to make sense of the spectrum of an unbounded operator
I'm trying to work my way up through various definitions in order to understand the formulation of the spectral theorem for unbounded operators, in which figure projection valued measures. I'm having ...
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Continuous function of a projection operator is itself?
Let $\{\psi_{n}\}_{n\in \mathbb{N}}$ a sequence of unit elements of a given Hilbert space $\mathscr{H}$. These are not necessarily orthogonal. For each of these vectors, define the projection operator ...
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Infimum of numerical range = infimum of spectrum?
Let $\mathcal{H}$ be a Hilbert space and $A: D(A) \to \mathcal{H}$ a densely defined self-adjoint operator. Its numerical range is defined to be the set $\{\langle x, Ax\rangle: \|x\| =1\}$. My ...
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Difference on two versions of spectrum of linear bounded operator
I have seen two different kinds of definitions on spectrum, one is from Wiki and another one can also be found on many materials.
Suppose that $A$ is a bounded linear operator on a Hilbert space $\...
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Compactness nilpotent operator [duplicate]
Let $A \in L(X) $, where $X$ is Banach space, and given $A^2 = 0$ then A is compact operator.
I found out spectrum of $A$ is zero, but I still can't understand, what should I do next.
Can someone help ...
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Interval in AC spectrum of self-adjoint operator
This may be an obvious and silly question, but I'll ask anyway. I was wondering whether a spectrum of a self-adjoint operator with non-trivial AC part can be totally disconnected.
I know that the ...
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How to find the eigenvalues of the operator?
Find the eigenvalues of the operator $(Ax)(t) = \frac{1}{t^\alpha} \int_{0}^{t} s^{\alpha - 1} x(s) \, ds$ , $Re\alpha>\frac{1}{p}$, in $L^p[0,1]$ space.
I tried $(A-\lambda I)x(t)=\frac{1}{t^\...
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Discrete spectrum of laplacian $ \sigma_{disc}(\Delta)$ on $ H^2(\mathbb{R}^n)$?
Let $\Delta: H^2(\mathbb{R}^n)\subseteq L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$
be the Laplace operator in the weak sense.
What is the discrete spectrum of $\Delta$?
the discrete spectrum ...
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Integral with respect to a projection valued measure?
If $H$ is a infinit dimensional seperable Hilbert space and $A:D(A)\subset H \rightarrow H$ a possibly unbounded selfadjoint operator. Then it is said that
$$A=\int\limits_{\sigma(A)} \lambda dP_{\...
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Do these properties hold for trace class operators?
I want to check two properties of trace class operators which I believe to be true an even have some skectches of proofs. In what follows, $A: D(A) \to \mathscr{H}$ is a densely defined self-adjoint ...
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Spectrum of a random matrix with distinct distributions per row
Let $r(t)$ a Gaussian function with maximum $r_{\max} = r(\frac{n}{2})$ for $n \in \mathbb{N}$. Let $\sigma$ denote the amplitude of the Gaussian curve, so that higher $\sigma$ values imply a slower ...
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Unitary operator in the spectral theorem is unique?
Assume $H$ is a Hilbert space and $A=A^*\in L(H)$. Let $x$ be a cyclic vector of $A$. Then there is a unitary operator $U$ from $L^2(\sigma(A), E_x)$ to $H$, where $E_x$ is the spectral measure of $A$....
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Minimization problem of the Fiedler vector
I am currently stuyding spectral graph theory and in my slides I stumbled on the following solution to obtain the Fiedler vector:
\begin{align}
\min_x x^T L_G x \\
\text{s.t. } x^T x = 1 \\
x^T \vec{...
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what is $J$-class set of zero in semigroup $\{e^{tS}\}_{t\geq 0}$ where $\sigma(e^S)\subseteq \mathbb{C}\setminus \overline{D}$
Let $S:X\to X$ be a bounded operator with $\sigma(S)\subseteq \{z\in\mathbb{C}: Re(z)>0\}$. Define $T(t)= e^{tS}$ and $\mathcal{T}=\{T(t)\}_{t\geq 0}$.
Define $$D_\mathcal{T}(x)= \{y: \text{ there ...
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Intuition behind the spectral theorem in infinite dimensions
I would like to verify my intuition behind the spectral theorem in infinite dimensions. For the moment I am putting bounded/unbounded and domain issues aside.
In finite dimensions the spectral theorem ...
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Proof of inequality using characteristic function
In part of some proof, I have a doubt in the following step:
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space. Let $M\in\mathcal{A}$, $(M_n)_{n\in\mathbb{N}}$ be a sequence of disjoint set of $\...
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Proof of multiplicativity of spectral measures
I'm reading the book "Introduction to Hilbert space and the Theory of Spectral Multiplicity" by Paul R. Halmos where I have trouble to follow the proof of Theorem 36.2.
Theorem 36.2
If $E$ ...
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Compact operator on Hilbert space: clarification of Wikipedia article
I have been trying to work through the Wikipedia article titled compact operator on a Hilbert space. I have made it though to the section 'Spectral theorem', subsection 'the idea'. This section seems ...
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Is $\sigma(-\Delta)=\sigma_{\mathrm{ess}}(-\Delta)$? Or under which conditions do we have this?
Let $\Delta: H^2(\mathbb{R}^n)\subseteq L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$
be the Laplace operator in the weak sense.
A Lemma in the book of Borthwick (Spectral Theory) says:
It is ...
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Borel functional calculus (Detail in construction)
In a course I've seen the proof of the following theorem:
Let $A\in\mathcal{L}(\mathcal{H})$ be a self-adjoint operator. Then there is a unique map
$\phi_{A}:\mathcal{B}(\sigma(T))\to\mathcal{L}(\...
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Spectral theorem for diagonal matrix in different inner product spaces
I learned a special case of the spectral theorem for finite dimensional inner product space. As I understand it states that a real matrix is orthogonally diagonalizable with real eigenvalues iff it ...
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Is $\|u\|_\infty\leq C\|u\|_{L^2}$ true for $u \in H^2(\mathbb{R}^{n})$ for $n>3$?
The Sobolev Lemma says if $u \in H^m(\mathbb{R}^n)$ $(m\in \mathbb{N})$, $k\in \mathbb{N}_0$ such that $k<m-n/2$ then $u\in C^k(\mathbb{R}^n)$ and for $|\alpha|\leq k$
$$\sup\limits_x |\partial^\...
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Spectrum and Resolvente
I have the following problem. Its supposed to be very easy but haven't yet properly digested the topic so i struggle.
Let H be be a separable complex Hilbert space with dim H $=\infty$, and let $A\in ...
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Proof of Multiplication version of spectral theorem for bounded normal Operators using the theorem for bounded self-adjoint operators
I would like to prove that for a normal operator T on a Hilbert space H there exists a measure space $(\Omega, \Sigma, \mu)$, a unitary operator $U: H \to L^2(\Omega)$ and a bounded measurable ...
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Is there a case in which PSD matrices $A$ and $B$ satisfy the condition $A \approx B$ but $A^2 \not \approx B^2$?
For two PSD matrices $A, B$ and a positive number $\alpha$, let's define $A \approx_{\alpha} B$ as $A \preceq \alpha B$ and $B \preceq \alpha A$, where $A \preceq B$ means $B - A$ is PSD. And if there ...
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