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

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Proof of the spectral theorem

I am currently going to through my proof of the spectral theorem that we had in class, but I feel that I have copied some nonsense from the board. So we defined the Cayley transform $U= ...
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Discrete Laplace: ONB

Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ Regard the unitary transformation: ...
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Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
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Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
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Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- ...
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Almost Mathieu operator

I'm having trouble showing that the almost Mathieu operator given by $$(Hu)_n = u_{n+1} + u_{n-1} + 2\lambda \cos \ [2 \pi (w + n\alpha)]u_n$$ Where $\lambda, \in \mathbb{R}$$, \alpha, w \in ...
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Resolvent: Definition

Given a Banach space. Consider linear operators: $$T:\mathcal{D}(T)\to E:\quad T(\kappa x+\lambda y)=\kappa T(x)+\lambda T(y)$$ (No other assumptions on the operator!) Denote for shorthand: ...
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The second smallest eigenvalue of a complete binary tree

Apparently it is true that the second smallest eigenvalue of a complete binary tree is $\theta(\frac{1}{n})$. Can someone point out a reference which proves this?
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What is the notion of “character” in the context of Cayley graphs?

I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to ...
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Can one go from eigenvectors (and eigenvalues) to the SVD of a matrix?

If I know all the eigenvectors and eigenvalues of a matrix, can I use that to directly write down the SVD of a matrix? (...of course one trouble is that for the $0$ eigenvalues of the matrix, the ...
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Relationship between spectrum of $-\Delta:H^1(M) \to H^{-1}(M)$ and $-\Delta:L^2(0,T;H^1(M)) \to L^2(0,T;H^{-1}(M))$?

Let us take a compact Riemannian manifold $M$. Let us define $-\Delta:H^1(M) \to H^{-1}(M)$ by $$\langle -\Delta u, v \rangle = \int_M \nabla u \nabla v$$ and $-\tilde \Delta:L^2(0,T;H^1(M)) \to ...
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Is this graph and its spectrum understood?

Consider the graph whose vertices are labelled by the binary representation of the integers from $0$ to $2^{d}-1$ for some $d \in \mathbb{N}$. So its a graph with $2^d$ vertices. An edge exists ...
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1answer
50 views

Unitary Equivalent of Derivative in Fourier Space

It is known that for $L^2(\mathbb R)$ the operator $Tf(x) = if'(x)$ is unitary equivalent to $\hat T \hat f(\xi )= \xi \hat f(\xi) $. Where domain of T is $H^1(\mathbb R)$. Hence the Spectrum of T in ...
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C* Algebra Positivity

STATEMENT: This is a proof from one of Qiaochu's notes on $C^*$ algebras. Proof: Let A be a $C^*$ algebra.We now want to show that for any $c\in A$ we have $c^*c\geq 0$. Suppose otherwise.We know ...
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28 views

C* algebra and ordering

It seems that given positive elements in a $C^*$ algebra A, we can give an ordering on its elements. Namely given elements $a,b\in A$ is positive iff $a\geq 0$ and $b\leq 0$ iff $-b\geq 0$. My ...
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Computation the different spectrums of an operator

Let $X:=C^0([0,2],\mathbb C)$,$\phi\in X$ and $T\in L(X)$ defined as: $$(Tf)(t):=\phi(t)f(t),t\in [0,2]$$ Compute: $\sigma_p(T),\sigma_c(T),\sigma_r(T),\sigma(T)$ and $\rho(T)$ I am quite new to ...
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Spectral mapping theorem of the measurable functional calculus

Let $H$ be a hilbert space and $T\in L(H)$ a self adjoint operator. Show that we have in general $\sigma(f(T))\neq f(\sigma(T))$ Any tips? If I choose a self adjoint operator how the measurable ...
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83 views

Selfadjoint Operator: Basic Criterion

For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way?
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34 views

Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
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Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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Is this some kind of Perron-Frobenius theorem?

If one has a matrix $A$ whose entries are all $\geq 0$ (or $>0$) then there apparently exists a diagonal matrix $D > 0$ (i.e all the diagonal entries are positive) and a constant $\alpha >0$ ...
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For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
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44 views

Weyl's law, meaning of the asymptotic formula, does it imply a bound?

Weyl's law states the eigenvalues of the Laplacian behave as $$\lambda_j \sim f(n)j^{\frac 2n}\quad\text{as $j \to \infty$}$$ where $n$ is the dimension. Does this literally mean that, $$\lim_{j \to ...
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Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
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interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
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Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...
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Spectrum of convolution operator

I was trying to find the spectrum of the convolution operator $$ J \ast u = \int_D J(x-y) u(y) dy $$ for bounded domain $D \subset \mathbb{R}$. Does anybody know it or have a reference for me? ...
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Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
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Modifications to the definition of graph Laplacian?

Many people have defined various definitions for graph Laplacian. For example see here [1]. What is common between various definitions of Laplacian that makes all of them ``Laplacian"? For example ...
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Eigenvalues of compact operator don't have nonzero accumulation points

In the book Elements of the theory of functions and functional analysis of Kolmogorov and Fomin, there is a proof of the following theorem, Every compact operator $A$ on a Banach space $E$ ...
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Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
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Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
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Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
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Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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Every normal operator on a separable Hilbert space has a square root that commutes with it

Show that every normal operator on a separable Hilbert space has a square root that commutes with it. Uniqueness? My attempt: Let $T$ be a normal operator. By polar decomposition $T=U|T|$ where ...
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Image of a projection

Show that $\lambda$ is an eigenvalue for normal bounded linear operator $N$ on Hilbert space $H$ with spectral measure $E$ iff $E(\{\lambda\})\neq 0$, in which case the range of $E(\{\lambda\})$ is ...
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Finding an explicit formula for $\int_\Gamma (z - A)^{-1} dz$, where $A$ is an operator on $l^p$

I am working on an example from the spectral theory of linear operators on Banach spaces. This is example 6.10 from chapter VII of Conway's A Course in Functional Analysis. Let $\{\alpha_n ...
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Self adjoint operator has spectrum around 0 and 1

I need to prove the following statement: Let $\mathcal{A}$ be a unital $C^*-$Algebra, $A$ a self-adjoint element and $P$ a projection, so $P^2=P=P^*$. Let $\delta :=\|P-A\|$. I want to prove that ...
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Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
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Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
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Dimension of generalized eigenspace.

Let $T \in \text{Hom}_F(V,V)$, suppose the characteristic polynomial of $T$, $c_T(x) = (x- \lambda)^kp(x)$, where $p(\lambda) \neq 0$, show that $\text{dim}_F (E_{\lambda}^\infty) = k$, where ...
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Special Elements: Spectrum

Given a C*-algebra with unit $1\in\mathcal{A}$. For normal elements one has: $$A^*=A^{-1}\iff\sigma(A)\subseteq\mathbb{S}$$ $$A^*=A\iff\sigma(A)\subseteq\mathbb{R}$$ ...