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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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On computing multiplicity function for self adjoint operator with nonatomic spectral measure

Suppose $T$ is a self-adjoint operator in $B(\mathcal{H})$ with $\sigma(T)$ is spectrum of $T$. $\mu$ is a spectral measure. For the operators having general continuous spectrum how to calculate the ...
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Spectral theory: Operator compact implies existence of convergent subsequence

So, I'm looking at the proof of the spectral theorem for self-adjoint compact operators in my functional analysis lecture notes (an introductionary class). We defined a compact operator $T$ as a ...
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Relationship between spectral projection and spectral measure

Let $H$ be a Hilbert space and $T \in \mathcal{L}(H)$ be a normal operator. It is known, that there exists a spectral measure $E$ on $(\mathbb{C},\mathcal{B})$ (where $\mathcal{B}$ is the set of Borel ...
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eigenfunctions of laplacian on infinite strip

I am trying to solve \begin{equation} -\Delta{u} = \lambda \cdot u \end{equation} with zero-boundary-conditions on the unbounded domain $\Omega = \mathbb{R} \times (0,\pi)$ for $\lambda \ge 0$ in the ...
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Approximate graph

Let $L_{G}$ be the Laplacian of a graph $G$ with irrational eigenvalues. I am curious to know: Is there any efficient way to find an approximate graph $\hat{G}$ such that all the eigenvalues of this ...
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30 views

Show that an operator with certain properties is an isomorphism

Suppose $H$ is an infinite dimensional separable Hilbert space, with $\langle \cdot,\cdot\rangle$ denoting the duality pairing between the dual $H'$ and $H$ and $(\cdot,\cdot)$ the inner product on $...
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Regarding spectral multiplicity

Suppose $T$ is bounded self-adjoint operator on a Hilbert space $\mathcal{H}$ which has no spectral multiplicity means multiplicity function is constant 1. Let $T$ has no eigenvalues. If $\lambda$ is ...
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Some questions about the spectral composition of a nonnegative self-adjoint operator

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
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31 views

Non-diagonalizable compact operators and the trace class condition

For a compact operator $A$ on a Hilbert space, it is said that $A$ is trace class if, for some (and hence any) orthonormal basis $\{e_n\}_{n \in \mathbb{N}}$, the series $$ s_k == \sum_{n=1}^k \left&...
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39 views

On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
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On spectral multiplicity

How direct integral decomposition of self-adjoint operator in $B(\mathcal{H})$ is connected with multiplicity? If we take $M_{f}$ on $L^{2}(X,\mu)$ what is it spectral multiplicity? How is direct ...
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the spectrum of a bounded linear operator on $X\times X$

If we consider $X\neq\{0\}$ to be a complex Banach space then the product $X\times X$ is a Banach space with the norm $\|(x,y)\|=\|x\|+\|y\|$. $T(x,y)=(x + y,x - y)$ is then a bounded linear operator ...
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31 views

Source for a particular proof of the spectral theorem.

Consider the following ''spectral decomposition'' for self-adjoint compact operators: If $T\neq 0$ is a self-adjoint compact operator on a Hilbert space $H$, then there exists a sequence $\{\...
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37 views

Three Questions about a weighted right shift operator.

Let $T:l^{2}\rightarrow l^{2}$ be a bounded operator defined as follows, If $x=\left( x_{1},x_{2},...\right) $ and $Tx=\left( y_{1},y_{2},...\right) $ then $y_{n}=0 \text{ if }n=1 $ $\text{ }=\frac{...
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Diagonalize the cyclic shift operator

Diagonalize this nxn matrix $\begin{bmatrix} 0&1&0&&&...&&&0\\ 0&0&1\\ &&0&1\\ &&&0&1\\ &&&&0&.\\ &...
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51 views

A self-adjoint operator with essential spectrum={0} is compact

Does every self adjoint operator (on a Hilbert space) with essential spectrum={0} is a compact operator ?
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32 views

A self-adjoint operator without eigenvalues and with spectrum equal to {0}

Let $A$ be a self-adjoint operator on a Hilbert space $H$. We know that the spectrum of $A$ ( $\sigma(A)$) can be decomposed into an essential spectrum ($\sigma_{ess}(A)$) and a set of eigenvalues ($\...
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Von Neumann ergodic theorem for purely continuous spectrum

$\newcommand{\1}{1\negthickspace{\mathrm{I}}}$ Von Neumann ergodic theorem states that, if $U(t)$ is a one-parameter group of unitaries acting on a Hilbert space $\mathcal{H}$, we have $$ \lim_{T\to ...
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Invertible elements in $l^1(\mathbb Z)$

The vector space $l^1(\mathbb Z)$ with $||x|| = \sum_{n \in \mathbb Z} |x_n|$ and $x * y(t) = \sum_{k \in \mathbb Z} x(k)y(t-k)$ forms a unital complex Banach algebra, with the unit being $\mathbf 1(0)...
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40 views

RAGE theorem for absolutely continuous spectrum?

Given a (possibly unbounded) self adjoint operator $A$ acting on a Hilbert space $\mathcal{H}$, the RAGE theorem gives a characterisation of $$ \|P_{c}\psi\|^2 $$ where $\psi\in\mathcal{H}$ and $P_c$ ...
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Construct operator with given spectrum. [duplicate]

The Spectrum of a bounded operator on a Banach space $X$ is always a compact subset of $\mathbb{C}$. What about the converse? Given any compact subset $K \subset \mathbb{C}$ is it always possible to ...
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Show that this second type Fredholm equation doesn't admit a solution using fredholm theorems

Given the following fredholm integral equation $g(s) =f(s)+\lambda\int_{0}^{2 \pi}sin(t+s) g(t) dt$ defined from $C[0,2\pi]$ over itself. Show that if $f(s) =s$ then the equation doesn't have ...
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Spectrum of Derivative Operator

Good evening! Given the operator $A$ acting on $L^2(0,1)$ with $Au = u'$, $\mathcal{D}(A) = \{u \in H^1(0,1):u(1)=0\}$, I am trying to show that $A$'s spectrum is empty, which is stated as easy to ...
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1answer
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Find projection-valued measure associated with parity operator

Let's define parity operator as follows: $$\pi:L^2(\mathbb{R})\to L^2(\mathbb{R})$$ $$\psi(x)\mapsto \psi(-x)$$ It's easy to show that $\pi$ is a self-adjoint operator and its spectrum is just $\sigma(...
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“bounding” an unbounded operator

I was wondering if, given a certain unbounded operator on a Hilbert space, it can (naively speaking) be "cutted" (or "bounded") by certain projections. So, thinking about this in a more sensible way, ...
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1answer
38 views

Converse of the Spectral Theorem

It is known that the spectrum of a compact operator $T \in B(X)$, where $X$ is an infinite-dimensional Banach space, is given by $\sigma(T)=\sigma_p(T) \cup \{0\}$ and $0$ is the only accumulation ...
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1answer
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Equivalence of definitions for the approximate point spectrum

Let $T: X \rightarrow X$ be a continuous, linear operator on some Banach space $X$. We defined the approximate point spectrum $AP\sigma(T)$ as the set $$ \{ \lambda \in \mathbb{C} : \lambda - T \;\...
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37 views

Sequence $(u_n)$ such that

Consider the following operator $H=\partial_{x^2}+x^2$ acting on $L^2(\Bbb{R})$. It's well known that the spectrum of $H$ is $\Bbb{R}$. So by weyl's criteron there exists $(u_n)\in D(H)$ such that $||...
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1answer
37 views

Possible spectra of singular Sturm-Liouville problems

A Sturm–Liouville (SL) eigenvalue problem with separated boundary condition on $[a,b]$ $$(py')'-qy=-\lambda^2wy$$ is regular if $p(x),w(x)>0$ and $p(x),p'(x),q(x),w(x)$ are continuous in the finite ...
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Prove that there is no self-adjoint extension using deficiency indices

Consider an operator $P =-i\frac{d}{dx} : dom(P) \to L^2(\mathbb{R}^+)$ where $$ dom(P) = \{ f \in \mathcal{D}(\mathbb{R}^+) : f(0)=0\}$$ where $\mathcal{D}(\mathbb{R}^+)$ - smooth compactly ...
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1answer
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Self-adjoint element of $C^*$-algebra has real spectrum

I'm trying to understand the proof in $\S$3.9 on p.23 of these notes: http://strung.me/karen/CStarIntroDraft.pdf. The argument is as follows: Let $A$ be a unital $C^*$-algebra, let $a \in A$ be self-...
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47 views

Spectral Theorem for Unitary Operator

It is well known that the following - in many literature - called the Spectral Theorem for Unitary Operator. I would like to know where i can find further information about it and its proof.
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Resolvent estimate of compact perturbation of self-adjoint operator

Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \...
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Reference for spectral theorem for normal operators.

Does anyone know of a good reference for the spectral theorem (projection valued measure version) for possibly unbounded normal operators? I would also be interested in examples where this sort of ...
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1answer
27 views

When does a matrix have only positive eigenvalues?

When does a matrix has only positive eigenvalues? I know that you can say if the eigenvalues are real or not by saying if the matrix is selfadjoint or skewadjoint, but how can you prove that has ...
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1answer
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what is Matrix of a linear transformation?

I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as $$T=\begin{pmatrix}2&-3\\3&2\end{...
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Limit circle/point of ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem $$(x^2y')'-[(x/2+a)^2+a]y=-\lambda^2y(x),$$ defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ and parameter $a>0$. ...
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Bounded function of compact normal operator on Hilbert space is normal

Let $H$ be a Hilbert space and consider a compact normal linear operator $A:H \to H$. Moreover, let $f$ be a bounded function on the spectrum $\sigma(A)$ of $A$ and consider the operator $f(A)$ in the ...
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Are projection valued measures continuous?

Let $U_t$ be a one parameter subgroup of normal bounded operators on a complex Hilbert space $H$. For each $t\in \mathbb{R}$, $U_t$ defines on the Borel subsets of $\mathbb{C}$ a projection valued ...
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1answer
33 views

When does the spectrum of an element in a Banach algebra with involution lie in the open right half-plane?

Let $A$ be a Banach algebra with involution, $x\in A$ and $t\in {\mathbb R}$ such that $t>\rho(xx^*)$. Show that $\sigma(te-xx^*)$ lies in the open right half-plane. I have no idea! It's obvious ...
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35 views

Bounding operator norm of matrix using eigen-values

I have a symmetric matrix $A$ and I have managed to show that all eigenvalues of $A$ lie in $[-c,c]$. I want to prove that the operator norm of $A$ is less than $c$ i.e. $||A||\leq c$. I could prove ...
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Find point spectrum and spectrum of integral operator

Let $A:L^2(0,\pi) \to L^2(0,\pi)$ be defined by $(Af)(x)=\displaystyle\int_{0}^\pi \sin(x-y)f(y)dy$. Find the point spectrum and spectrum of $A$. I am not sure how to go about this. I thought to ...
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Kernel of adjoint and cokernel of operator

Let $D$ be a Fredholm operator and $D^\dagger$ is its adjoint. Is the dimension of $ker ~ D^\dagger$ equal to the dimension of $coker ~D$ ? If so, can someone sketch the proof?
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Approximate point spectrum of multiplication operator in $L^p$

Let $M_\phi f=\phi f, \,\, \phi\in L^\infty(X,\Omega,\mu),\,\, f\in L^p(X,\Omega,\mu),\, 1\leqslant p \leqslant \infty.$ I finded $$\sigma(M_\phi)=\left\{ \lambda \in \mathbb{C} \mid \not \exists \...
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Does convergence in Hilbert-Schmidt norm imply convergence of singular values?

Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = \sum_n \lambda_n u_n \otimes v_n$. Now let $A_i$ be a sequence of operators ...
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1answer
34 views

Concavity of $\log u(x)$

Let $u>0$ be a solution of the following equation $$\begin{cases}-[|u'(x)|^{p-2}u'(x)]'=\lambda_1\cdot u(x)^{p-1}&,x \in (a,b)\\u(a)=u(b)=0 \end{cases},$$ where $\lambda_1$ is the minimum of ...
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2answers
65 views

Proving $\lambda_{\text{min}}\leq R(x)$ without Spectral Theorem

Given a real-symmetric (or Hermitian), positive definite matrix $A$, it is well known that: $$\lambda_{\min}\leq\dfrac{(x,Ax)}{(x,x)}. \tag{1}$$ This is a direct consequence of the min-max theorem ...
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Spectrum of translation operator in $L^2$

From exam preperation. I consider the operator $T f(x) := f(x − 1), x ∈ \mathbb{R}$. First on the space of all function $f:\mathbb{R}\rightarrow \mathbb{C}$. There I found that any number other than $...
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1answer
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Spectrum of Banach algebra with coordinate multiplication

Consider $X=l^p$ , $p \in [1, \infty )$. I proved that $X$ with coordinate multiplication is commutative Banach algebra without unit. I have got a problem to find the spectrum of general element of ...
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Connection between the eigenfunctions of the compact operators $T[f](x\in H_1)=\int_{H_1}k(x,y)f(y)dy$ and $R[f](x\in H_2)=\int_{H_1}k(x,y)f(y)dy$?

Let $H_1$ and $H_2$ be Hilbert spaces. Suppose we have a compact integral operator $T:H_1 \to H_1$ given by $$ T[f](x) = \int_{H_1} k(x,y)f(y)dy, \quad \quad x \in H_1. $$ Suppose we also have a ...