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

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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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Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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69 views

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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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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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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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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39 views

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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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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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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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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Application of the spectral mapping theorem

Let $T:L^2((0,2)\rightarrow L^2((0,2))$, $(Tx)(t):=\begin{cases} x(t+1), & 0<t<1\\ 0,& \text{elsewhere} \end{cases} $ Show that $T$ is well defined and $\sigma(T)=\sigma_p(T)=\{0\}$ ...
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93 views

Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f''$ ...
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Is there a diagonal matrix D such that DMD is SDD, where M is SPD matrix?

Let $M$ be symmetric and positive definite matrix (SPD). It is known [1] that if $M$ is SPD and in addition satisfies $M_{ij}\leq 0$, for $i\neq j$ (called M-matrix) then there is a positive ...
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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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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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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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82 views

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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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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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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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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Spectral Measures: Unitary Map [duplicate]

This thread is a record. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}\to\mathcal{H}:\quad N^*N=NN^*$$ and its spectral measure: ...
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69 views

Spectral Measures: Embedding

This thread is just a note! Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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129 views

Spectral Measures: Uniqueness

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E^{(\prime)}:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote their operators by: ...
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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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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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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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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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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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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 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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50 views

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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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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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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38 views

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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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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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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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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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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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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Distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
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36 views

Special Elements: Spectrum

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

Summary: Spectrum vs. Numerical Range

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Denote for shorthand: ...
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41 views

Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but its spectrum ...
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Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
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Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the ...