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

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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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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( ...
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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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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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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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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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32 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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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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40 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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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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62 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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28 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. ...
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35 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 ...
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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, ...
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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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27 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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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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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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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( ...
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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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27 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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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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22 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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54 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 ...
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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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29 views

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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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 ...
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82 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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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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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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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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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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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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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,}\,\,\, ...
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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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22 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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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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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\ ...
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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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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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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 ...
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Random matrix on Clifford algebra within a specific grade

There has been some discussions about random matrices on generic Clifford algebra arXiv:1312.6291. However I would like to consider a more specific case by restricting the random matrix within a ...
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Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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Detecting eigenvals of opposite sign

Consider a large (in the region of 500 by 500 up to 2000 by 2000) real, square symmetric matrix. What would be a good way of algorithmically determining if it has any pair of eigenvals with opposite ...
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Convergence of spectrum [closed]

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
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37 views

Spectrum of unbounded operators

I am currently a little bit confused. I am aware of a theorem that says that any closed and densely defined operator satisfies $\sigma(T^*)=\overline{\sigma(T)}.$ On the other hand, the operator ...