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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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How to prove Neumann series doesnt converge when spectral radius > 1?

For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$ If $r(T)>1$, I can show that series ...
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There is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$

Could someone give me a suggestion to solve the following problem problem? PROBLEM. Let $T : \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ and $\beta$ a basis of $\mathbb{R}^{3}$ such that $$ \left[ ...
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Proving that two variational problems are equivalent

Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \...
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Point spectrum of an integral operator

Let we have $$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy$$ so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
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Computation of the complex roots of the Laplace transform of a function?

I have a function $f \in L^1(\mathbb{R}_+, \mathbb{R})$ with Laplace transform $$ \forall \Re(z) \geq 0,~~ \hat{f}(z) := \int_{\mathbb{R}_+} { f(t) e^{-zt } dt}.$$ I know explicitly the expression of ...
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1answer
24 views

Relation between Schatten-$p$-norm and $l^p$ norm of operator matrix

Let $\mathcal H$ be a separable Hilbert space and let $(e_i)$ be some orthonormal basis. Let $K$ be a compact operator on $\mathcal H$ with matrix elements $K_{ij}=\langle K e_i,e_j\rangle$. My goal ...
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13 views

Given two commuting projection-valued measures, is there a product measure?

Let $H$ be a Hilbert space. Let $(\Omega_1,\mathcal A_1)$ and $(\Omega_2,\mathcal A_2)$ be measurable spaces. Let $$P_1: \mathcal A_1\to\mathfrak L(H), \quad P_2: \mathcal A_2\to\mathfrak L(H)$$ be ...
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24 views

Quadratic Forms Orthogonal Diagonalization Existence

Why does one assume that the eigenbasis for a quadratic form is orthogonal, hence orthogonal diagonalization. I understand that for hermitian and unitary maps one can show by spectral theorem an ...
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17 views

How do we derive the spectral projector associated with a simple eigenvalue?

Result 7.2.12 of Meyer's Matrix Analysis and Applied Linear Algebra gives the following: If $x$ and $y^*$ are respective right and left eigenvectors of a matrix $A$ associated with a simple ...
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Trouble understand proof of spectral theorem

I'm reading through this proof of the real spectral theorem. I don't understand the last line of "lucky fact 2" - why must $\overrightarrow{u}$ have been listed in the $v_{i}$?
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Periodic boundary conditions, general dimension, sets and spectral properties of $-\Delta$ - reference recommendation

Let's consider the eigenvalue problem $-\Delta u = \lambda u$ on the interval $[0,1]$ with periodic boundary conditions: $u(0)=u(1),$ $\frac{du}{dx}(0) = \frac{du}{dx}(1).$ Similar conditions could be ...
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42 views

Is there a way to concretely see cyclic sub-representations of the “Euclidean group” on $\mathbb Z$?

Let $\mu$ be a finite Borel measure on $S^1$. We have an action of $\mathbb Z$ on $L^2(S^1, \mu)$ defined by $n\cdot \varphi = e^{2\pi i n}\varphi$. The following is a standard theorem in functional ...
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16 views

Definition of spectral measures corresponding to koopman operator

Let $U$ be the koopman operator on $L^2(X,\mu)$ where $(X,T,\mu)$ is an MPT. We characterize the spectral measure by its fourier coefficients using bochner's theorem as $$\hat{\sigma_{f,g}}(-k) = \...
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operator with compact resolvent and spectrum

If the resolvent of an opertaor A is compact. The question is why does the spectrum and the ponctuel set becomes equal.That is, why $\sigma(A)$=$\sigma_{p}(A)$
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A Homogeneous Fredholm Equation of Second Kind

in my probability research I encounter the following integral equation for continuous non-negative $f: (0,\pi/4] \to \mathbb R$: $$ f(\varphi) = \int_0^{\pi/4} \frac {4} {\pi} \sin \varphi_0 \cos \...
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1answer
37 views

Help verifying the norm of the resolvent of a matrix

I'm reading a document where it is said that if $$A=\begin{pmatrix}0 & 1\\0 & 0 \end{pmatrix}$$ then the norm of the resolvent for $z \neq 0$ is given by $$\|R(z,A)\|= \frac{\sqrt{2}}{\sqrt{1+...
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17 views

Eigenfunctions of the operator $Au:=-u''+ix^2u$ in $L^2(\mathbb{R})$

I need to find the eigenfunctions of the operator $$Au:=-u''+ix^2u$$ if the domain of $A$ is the set of functions in $L^2(\mathbb{R})$ with absolutely continuos derivatives $u,u'$ in $L^2(\mathbb{R})$...
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37 views

Spectrum of $T(x_1, x_2, \ldots)=(\varepsilon x_1, x_1,x_2,\ldots)$ in $\ell^2$ for $\varepsilon>0$

Let $\varepsilon>0$ and $T:\ell^2\to \ell^2 $ the operator defined by $$T(x_1, x_2, \ldots)=(\varepsilon x_1, x_1,x_2,\ldots)$$ Can you help me to calculate the spectrum of $T$, please?. I ...
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36 views

Root of closed operator

prop:Let $(H,(,))$ be a Hilbert space and $T$ be a non-negative self-adjoint closed operator s.t., domain of $T$ is dense in $H$. Then, $\sqrt{T}$ is closed operator. By spectral decomposition, We ...
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19 views

On standard form of particular subalgebra of vN algebra

If $M$ is in standard form, consider the action of finite group on $M$, does the fixed point sub algebra under the action is in standard form?
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35 views

When is a surjective homomorphism between two unital Banach algebras bounded?

Let $A$ and $B$ be unital Banach algebras and $\theta : A\to B$ a surjective homomorphism between these two spaces. What is a sufficient requirement for $\theta$ being bounded, and how would a proof (...
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How spectral projection of a self adjoint operator behaves under conjugation of positive operator or unitary operator

Let $T$ be a self adjoint operator on $\mathcal{H}$, what are the relations of spectral projections of $UTU^{*}$ and $U$ , if U is unitary or positive?
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27 views

Is there any relation between $||(z-A)^{-1}||$ and $||(\frac{1}{z}-A^{-1})^{-1}||$ for $z \in \rho(A) \setminus \{0\}$?

Let $A: D(A) \to X$ be an operator defined in a Banach space $X$ which has bounded inverse. I know that if $z \in \rho(A) \setminus \{0\}$, then $\frac{1}{z} \in \rho(A^{-1})$. But, is there any ...
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17 views

Existence of fixed points for this Markov operator.

Perhaps math overflow is a better place to put this but I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular ...
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17 views

Can norm of resolvent of closed operator grow away from its spectrum(but near to its numerical range)

I hope my question is simple to answer but I could not find anything up to now. Given a closed densely defined operator $A \colon H \supseteq D(A) \to H$ in a Hilbert space $H$. For many types of ...
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1answer
25 views

what is the spectrum of a linear map and does 0 belong to it?

can somebody help me understand what the spectrum of a linear map is? from what i have seen its the set of lambdas such that A-I(lambda) is not invertible for a linear map A:X->X, so its a set which ...
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17 views

ascent of product of two linear operators

Let $S,T: X \rightarrow X$ be two bounded linear operators on a Banach space $X$. Let $ST=TS$. If asc$(ST$) is finite then is asc$(T)$ finite? For a bounded linear operator $A$ on a Banach space $X$ ...
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1answer
21 views

Polynomial evaluated on a Normal Bounded Linear Operator

Let X be a complex Hilbert Space and A be a Normal Bounded Linear Operator. Show that the radius of the spectrum of A is equal to the norm of A. Deduce that if P is a polynomial, then the norm of P(...
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1answer
23 views

Proving that any unitary matrix can be diagonalised by a similar matrix

I'm having struggles with understanding important facts about spectral theorem in finite dimensional spaces. For hermitian matrices, I saw in classes that the similarity matrix that diagonalises any ...
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Spectrum of an unbounded operator is not compact.

My problem: Let $X=C[0,\pi]$ and define an operator $T: D \to X$ where $$D= \{x \in X \mid x',x'' \in X\quad \text{and} \quad x(0)=x(\pi)=0\} $$defined by $T(x)=x''.$ Show that spectrum of $T$ ...
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the spectrum of operator with compact resolvent

If we have an unbounded self adjoint operator A, that has a compact resolvent. The question is why $\sigma(A)=\sigma_{p}(A)$, where $\sigma(A)$ is the spectrum and $\sigma_{p}(A)$ is the set of ...
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1answer
32 views

When do singular values occur in pairs?

When playing around with a certain matrix I'm studying I noticed something interesting. The top singular values seem to occur in pairs: ...
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1answer
22 views

About spectrum of a multiplication operator ON the Hilbert-Schmidt space

$\newcommand{\h}{\mathcal H}$$\newcommand{\tr}{\mbox{tr}}$ In brief : Treating every bounded linear operator of a Hilbert space, as a multiplication operator which is an element of $B_2(\h)$ (...
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what are differential forms in a unital algebra

In Bourbaki, Theorie Spectrale, Ch. 1, No.2 the following construction is made: we let A be a commutative unital algebra with unit 1 (think of bounded linear operators on a Hilbert space, please). ...
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121 views

Extending the spectral theorem for bounded self adjoint operators to bounded normal operators

I'm currently preparing for an exam in functional analysis, and I have a question about the extension of the spectral theorem for bounded self adjoint operators to bounded normal operators. Starting ...
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1answer
29 views

Spectrum of a self-adjoint compact operator

Given a self-adjoint compact operator $A$ on a (separable) Hilbert space, is it true that the spectrum of $A$ is equal to the closure of the set of eigenvalues of $A$, or in symbols $$ \sigma(A) = \...
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1answer
29 views

Relation between spectrum of an operator and its cut down

Let $T$ be a self-adjoint operator in $\mathcal{H}$ with spectrum $\sigma(T)$, Let $P$ be a projection in the commutant of $\text{vN}\{T\}$, the von Neumann algebra generated by $T$, question what is ...
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8 views

Spectral methods on semiring matrices

It is always very useful to have the spectral decomposition of a matrix or to know something interesting about eigenvalues and eigenvectors of a matrix of interest. In algebraic algorithms, one often ...
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63 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

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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For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
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Show that the operator associated to a spectral decomposition on a Hilbert space is self-adjoint

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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Is the approximate point spectrum simply the union of the essential and point spectra?

I'm refreshing functional analysis from "Hilbert space operators in quantum physics" by Blank, Exner and Havlíček (Springer, 2008). They define essential spectrum for any closed operator on a Hilbert ...
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Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...
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38 views

What are the approximate eigenvalues of the right shift operator $R$ on $\ell_\infty$

I have shown that the spectrum of $R=\{z\in C||z|\leq 1\}$. Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $\forall |z|=1$, $z$ is an approx. eigenvalue. However, ...
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When is the spectral radius of a submatrix strictly bounded by the original matrix.

Let $A$ be a matrix in $\mathbb R^{n\times n}$. Define an orthonormal basis $\{ w_1,w_2,\dots,w_n \}$ of $\mathbb R^n$. Consider a subset of $m$ basis vectors, say $\{ w_j | j\in \mathcal J\}$ and $\...
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15 views

Eigenvalue of algebraic multiplicity $m$ is a pole of the resolvent of order $m$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X)$ be a bounded linear operator. Suppose that for some isolated point $\lambda \in \sigma(T)$ and some $m \in \mathbb{N}$ we have $\ker(T-\lambda I)^m ...
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Spectrum of an operator defined by spectral integral

First of all I want to thank you for the help you provide on this website! Whenever I had a hard time understanding things in math I visited this website and (nearly) allways found a hint or a ...
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1answer
31 views

On self adjoint operator having discrete spectrum

If $T$ is a self adjoint operator in $\mathcal{H}$, having discrete spectrum, is it true that set of eigenvectors form an orthonormal basis for $\mathcal{H}$? Under which condition it will form onb ...
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Is the limit of the spectral radius the spectral radius of the limit?

Let $A$ be an unital Banach algebra, $x \in A$ and $(x_n)$ a sequence in $A$ converging to $x$. I want to show that $$ \lim\limits_n \rho (x_n) = \rho (x).$$ I can show that $$\limsup \rho(x_n) \leq \...
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1answer
45 views

How can point spectrum equal residual spectrum?

I came across a result in a paper(Proposition 2.2 on the second page in https://www.ams.org/journals/tran/1988-306-02/S0002-9947-1988-0933321-3/S0002-9947-1988-0933321-3.pdf) which states: If $X$ ...