Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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Spectrum of operator in $l_2$

Sorry guys, I have a problem with this exercise. Let T an operator in $l_2$ Hilbert space: $$ (\textrm{T}x)_1 = x_2 , $$ $$ (\textrm{T}x)_2 = 0 , $$ $$ (\textrm{T}x)_n = x_{n-1} - x_n $$...
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35 views

How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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References on the spectral theory of the Floquet problem

The Floquet problem is about the linear ordinary differential equation $$ \dot{\psi} = A(t) \psi. $$ Here $A(t)= A(t+T)$ is a periodic $n\times n $ matrix. Suppose $A(t) =-i H(t)$ with $H(t)$ being ...
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41 views

Proof: the resolvent operator is holomorphic.

I tried to prove that the resolvent operator $$\rho(A) \to \mathbb C,\space \lambda \mapsto R_{\lambda}(A):=(\lambda id_X -A)^{-1} $$ is holomorphic, where here $A$ is a bounded linear operator from ...
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30 views

Nonempty intersection between approximate point spectrum and residual spectrum

On the Wikipedia page on "Spectrum (functional analysis)", it is mentioned that the approximate point spectrum and residual spectrum are not necessarily disjoint. Is there a straightforward example to ...
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35 views

Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...
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52 views

Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$

I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$) $$T((a_j))=\left( \sum_{j=2}^{\...
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5 views

Classify sub $C^*-$ algebras of $\mathbb{C}^{2 \times 2}$

Apparently if $A$ is a sub $C^*-$ algebra of the complex $n \times n$ matrices then we can characterize these subalgebras as block matrices.Now, for the case $n=2$ I was wondering if there is an easy ...
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1answer
39 views

Uniform convergence in Mercer Theorem for bounded kernels

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$ ...
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1answer
41 views

Orthogonal projection onto the eigenspace of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all ...
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34 views

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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1answer
33 views

The topology of the spectrum of a linear operator

In general a spectrum of a linear operator has a decomposition into three parts: point spectrum, continuous spectrum and residual spectrum. What I'm interested in is the topology of these parts of ...
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2answers
38 views

Clarification on point spectrum of an operator

From my understanding, if $\lambda$ is in the point spectrum, then $\lambda$ is a complex number such that it satisfies the equation $(T - \lambda I) x = 0$. My confusion arises from problems like ...
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52 views

Isolated Eigenvalues on the Extensions.

I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ...
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1answer
42 views

$\lambda$ is an eigenvalue iff spectral measure of $\lambda$ is nonzero

Let $M$ be a normal operator on a Hilbert space and let $E$ be the spectral measure of $\sigma(M)$ (the spectrum of $M$). Show that $\lambda$ is an eigenvalue to $M$ $\iff E(\{\lambda\})\not = 0$. ...
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1answer
46 views

Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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67 views

Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
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1answer
41 views

The k-th derivative of the resolvent set

I want to prove $$\frac{d^{k}}{dz^{k}}(zI-A)^{-1}=(-1)^{k}k!(zI-A)^{-k-1}$$ I have the resolvent equation $(zI-A)^{-1}-(\lambda I-A)^{1}=(\lambda-z)(zI-A)^{-1}(\lambda I-A)^{-1}$, i.e. $$\begin{...
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1answer
38 views

Spectrum of the laplacian on a Banach space

Is the spectrum of the laplacian on $L^1(0,1)$ with Neumann boundary conditions known?
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37 views

Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on $\...
2
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1answer
25 views

Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$

Let $V$ be a complex inner product space. Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$? If not, ...
2
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1answer
78 views

If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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32 views

Spectrum radius upper bound of Hadamard product

I am doing some research that relies on Hadamard product of two matrices bound, the most famous one that I encounter is : $\rho(A\circ B)=\rho(A)\rho(B)$ this seems to be trivial when I test it with ...
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1answer
55 views

Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
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31 views

What does the spectrum of the adjacency matrix of a graph tell you? [duplicate]

I am trying to search for an answer to the following question and I cannot find a straightforward answer. What does the spectrum of the adjacency matrix (set of eigenvalues and their multiplicities) ...
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1answer
17 views

T be compact operator defined on $L^2(\Omega)$ Show that the null space of T-I satisfies

Can anyone help me out on this one Let $$ \Omega\subset\text{R^d be a domain K}\,\in\,L^2(\Omega X \Omega)$$ and T be compact operator defined on $L^2(\Omega)$ by Tf(x)=$\int_\Omega K( x,y)\text{f(...
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How to do spectral decomposition?

I missed the last couple classes due to a family emergency and am trying to catch up with review questions. However, I can't seem to find an online source that teaches how to compute a spectral ...
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29 views

compactness in $\ell^2$

How can I show T is compact when T is defined as $$ \text{T :}\,\ell^2 \to\ell^2\,\text{by Tx=y where} \,y_j=\alpha_jx_j\text{and}\,\alpha_j\to0\,\text{as}\,n\to\infty$$
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33 views

Intuitive understanding of quantum ergodicity of eigenfunctions

I recently heard a talk on differential geometry where the speaker was using a result called quantum ergodicity of eigenfunctions. I am trying to see if I am getting the gist of the result correctly. ...
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24 views

Spectral Theorem for compact Operators

I think about the spectral theorem for compact operators on a Banach Space. And I come to a question: Can the Theorem be generalized to any Normed Space or a bigger subclass of TVS
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1answer
58 views

Eigenvalues of an integral operator

The following operator is defined on $L_2(0,1)$: $$Kf(t)=\int_0^1|s-t|f(s)ds$$ I am wondering how I can calculate the eigenvalues and eigenfunctions of such an operator. I start with $\int_0^1|s-t|f(...
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47 views

Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ...
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1answer
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Confusion on Theorem in Kato's book

On page 432 (pdf-page: 455) of Kato's book perturbation theory of linear operators, I do not understand why in Theorem 1.15 $$H_n = \int dE_n(\lambda)$$ instead of the ususal thing $$H_n=\int \...
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55 views

Laplace-Beltrami operator

I'm interesting in the Laplace-Beltrami operator on a sphere, more precisely its spectral properties including the spectral function, etc. So if someone can give me some references that treats this ...
2
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1answer
94 views

Spectrum of nonnegative operator

Let $A$ be a bounded, nonnegative operator on a complex Hilbert space $H$. Prove that the spectrum $$\sigma(A)\subset[0,+\infty].$$ We say that an operator $A$ is nonnegative if it is self adjoint and ...
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32 views

What theorem is this in the infinite dimensional case?

The spectral theorem for compact, self-adjoint operators is as I have understood the infinite dimensional case for orthogonal diagonalisation of a symmetric case in linear algebra? But in linear ...
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1answer
50 views

Problem involving the Spectral Mapping theorem.

Consider the following problem: Let $T$ be a bounded operator in a Banach space $X$. Use the Spectral Mapping theorem to show that $|\lambda^n|\le\|T^n\|$ for all $\lambda\in\sigma(T).$ Here's ...
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2answers
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Difference between the spectrum and point spectrum of an operator.

I have the following two definitions in my notes: The spectrum of an operator: We define $\sigma(T)$, the spectrum of T, by, $$\sigma(T):=\{\lambda\in\mathbb C: T-\lambda I\,\, \text{is not ...
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1answer
25 views

Computing spectra in Banach algebras

In general, computing the spectrum of a specific element in a Banach algebra can be very difficult. What are some of the less obvious tricks that you've encountered?
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1answer
29 views

How to find spectrum of a convolution operator

Say $k$ be s.t. $\hat{k}$ is a bounded function on an LCA group $G$ and $Tf=f*k$. Then $T$ is bounded on $L^2(G)$. Is there anything I can say about $\sigma(T)$? (except the properties that follow ...
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Holomorphic Functional calculus & approximate point spectrum

Let $A\in B\mathbb{(H)}$ and $f\in Hol(A)$,then $f({\sigma}_{ap}(A))={\sigma}_{ap}(f(A))$. I only know how to prove $f({\sigma}_{ap}(A))\subset{\sigma}_{ap}(f(A))$,but have no idea on the other side. ...
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Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
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1answer
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Find the eigenvalues of the operator T.

I have the following problem, "Suppose that $X=\ell^1$ and define the operator $T\in B(X)$ as follows: $$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, x=(x_1,x_2,x_3,.....
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4answers
226 views

Left Shift Operator Spectrum

Consider the Hilbert space $\mathcal{H}=l^2(\mathbb{Z})$ and define the left shift operator $\mathcal{L}:\mathcal{H} \rightarrow \mathcal{H}$ by $$ \mathcal L (a_n) = (b_n) \qquad \text{ where } \...
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1answer
23 views

Let $f$ and $g$ linear operators where $f$ and $g$ commute and $f$ has simple spectrumm, then there is $P$ a polynomial such thah $g=P(f)$.

Let $f : \mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ be a linear operator with a simple spectrum, furthermore, let $g : \mathbb{C}^{n}\rightarrow \mathbb{C}^n $ be a linear operator such that $f$ and $...
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1answer
59 views

Estimate spectral radius of operator product

In my research problem, I have to estimate the spectral radius of the following operator $\chi A$ where $\chi$ is a scalar function taking values 0 or 1 and $A$ is an operator. I can compute ...
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1answer
26 views

Are these critical points minima to the variational problem?

Let $\Omega\equiv (0, 1)\times(0, 1)\subset\mathbb{R}^2$ and consider the variational integral \begin{equation*} I[u]\equiv\int_{\Omega}\frac{1}{2}|Du|^2\ \mathrm{d}x-\frac{5\pi^2}{2}|u|^2\ \mathrm{d}...
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1answer
45 views

Topological characterization of the range of a bounded normal operator

Let $T$ be a bounded normal operator on a Hilbert space $H$. I want to prove the following statement: $\text{ran}(T)$ is closed if and only if 0 is not a limit point of $\sigma(T)$. I tried to use the ...
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1answer
19 views

Spectral Radius of A+B

I am solving a physical problem numerically which gives three real, symmetric and positive semi-definite matrices: $A$, $A_1$, and $A_2$; where $A=A_1+A_2$. I know that the following identities ...
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1answer
48 views

Show self-adjointness elementary

Is anybody aware of an elementary proof that $T^*T$ is self-adjoint where $T$ is closed and densely-defined? All proofs I found so far use the Friedrich's extension or other more sophisticated ...