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

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Finding an eigenvalue

I got the following statement to prove: Let $A \in \mathbb R^{n \times n}$ be a (column-)stochastic matrix, i.e. $A \geq 0$ (which means $a_{ij} \geq 0$ $\forall i, j \in \{1, \dots ,n\}$) and ...
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Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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8 views

Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition

Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one?
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39 views

Riemann mapping theorem - Personal project

I would like to work on the Riemann mapping theorem this summer. Does anyone could give me some good references linked to this objective. For your information, I now currently finishing a degree in ...
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46 views

Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for ...
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18 views

separable weakly dense subalgebra of a von Neumann algebra

This is an exercise of "A short course on spectral theory" by William Arveson. For any von Neumann algebra $\mathcal{M}$ on a Hilbert space $H$, I need to show there exists a unital ...
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20 views

Limit of sesquilinear forms is a sesquilinear form

Suppose $P_n$ is a monotone sequences of orthogonal projections in a complex Hilbert space $\mathcal{H}$. I want to show that the limit of the sesquilinear forms defined by: $\Gamma_n(x,y)=(x,P_n y)$ ...
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Monotone sequence of orthogonal projections on a complex Hilbert space

Suppose $P_n$ is a monotone sequence of orthogonal projections on a complex Hilbert space $\mathcal{H}$, i.e. $V_n= Im(P_n)$ is a decreasing or increasing sequence of subspaces and $P_n^\star=P_n$ and ...
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45 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
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54 views

Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all ...
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16 views

An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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50 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
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30 views

Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
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28 views

The Hilbert-Schmidt Theorem for Compact, Self-adjoint operators

Suppose $T$ is a compact, self-adjoint operator on a Hilbert space $\mathcal{H}$. Then there exists an orthonormal set $\{e_n\}_{n=1}^{\infty}$ of eigenvalues of $T$ such that every $x \in ...
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Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f'' $ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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Article in spectral theory

Can someone please help me to understand how we prove the first inequality (Page 7) in this article http://arxiv.org/pdf/1510.01567.pdf It seems that we use the Young inequality but i didn't know ...
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22 views

Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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Two formula for an operator

we define the operator P with domain $C_0^\infty(R^{2n})$ by $P=y.\partial _x -\partial _x V(x).\partial_y-\Delta_y+\frac{|y|^2}{4}-\frac{n}{2}$ I want to prove that $P=X_0 +\sum _{i=1}{^n}X_j^*X_j$ ...
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29 views

About the Mercer's theorem.

Doesn't the the Mercer's theorem say something stronger than just the spectral theory of compact self-adjoint operators on a Hilbert space applied to the reproducing "kernel" function? As in if I ...
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Inequality for an operator

We consider the operator $P$ with domain $C_0^\infty(R^{2n})$ defined by $P=y.\partial _x -\partial _x V(x).\partial_y-\Delta_y+\frac{|y|^2}{4}-\frac{n}{2}=X_0 +\sum _{i=1}{^n}X_j^*X_j$ where ...
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Problem with Spectral Theorem Proof

Claim: Let A $\in \mathbb{R}^{n \times n }$ be symmetric. Then there is an orthonormal basis of $\mathbb{R}^n$ consisting of eigenvectors of A. Sketch of proof: Induction on n. Claim is clear for ...
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The density of $C^1[0,2\pi]$

I am not sure if the inclusion $\{f \in AC[0,2\pi]: f(0)=f(2\pi)=0\}\subseteq \overline{\{f \in C^1[0,2\pi]: f(0)=f(2\pi)=0\}}.$ Here $C^1[0,2\pi]$ is the set of continuously differentiable functions ...
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29 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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18 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
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28 views

L2 Norm: Unfamiliar notation

In this article that I am reading, I am given a non-negative spectral function $w(\lambda)$ which is "interpreted as a weight function determining the scalar product of two functions $f(\lambda)$ and ...
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Spectral measure of a stationary time series

Let $(Z_t)$ be white noise with $E[Z_t^2]=1$ and $A$ and $B$ random variables such that $E[A] = E[B] = 0$, $E[A^2] = E[B^2] = 1$, $A$, $B$ and the infinite sequence $(Z_t)$ are independent ($(Z_t)$ ...
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Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by ...
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Spectral density of a filtered stationary process

I have a stationary time series $(X_t)$ with spectral density $f_X$, i.e. $$f_X(\lambda) = \frac{1}{2\pi}\sum_{h\in\mathbb{Z}} e^{-ih\lambda}\gamma_X(h)$$ where $\lambda \in (-\pi,\pi]$ and ...
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If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ...
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Change of variables of a polynomial

Can someone help me please to solve this problem: we consider $V(Y)$ a polynomial in $ R[Y_1,Y_2,..,Y_d] $ I want to prove that there existe an affine change of variables $ Y=AX+B,X=(X_1,X_2,..,X_d)$ ...
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Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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If spectral radius $\rho(A)<1$ , does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-||A||_{2})$ hold true?

If spectral radius $\rho(A)<1$, does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-\||A||_{2})$ hold true? If it is correct can somebody give me link to the proof for this inequality?
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ODE system with lower triangular coefficient matrix

Let $v_k$ satisfies the following equation $\frac{d}{dt} v_k(t)=\sum_{i=0}^{k-1}v_k(t)$ with initial data $v_0(t=0)=1$ and $v_j(t=0)=0$. Notice that the coefficient matrix $A$ of this ODE system ...
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Formula connecting the resolvent opeartor andthe spectral density?

I want to know if it is a formula connecting the resolvent opeartor $(\lambda - T)^{-1}$ for a selft-adjoint operator $T$ and its spectral density $e_{\lambda}$. Thank you in advance
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Formula connecting the resolvent and the heat kernels

Using the well known formula connecting the resolvent and the heat operators associated to a selft-adjoint opeartor $A$ \begin{align} (\zeta - A)^{-1} = \int_{0}^{\infty} e^{-\xi t} \, e^{t A} dt; ...
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Direct Integral: Scalars

Given a Borel space $\Omega$. Regard the Hilbert Space: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mathcal{H}:=\mathcal{L}^2(\Omega;\mu)$$ Denote the Borel Projections: ...
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21 views

Largest element in inverse of a positive definite symmetric matrix.

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
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spectrum of unbounded self-adjoint operators

I'm self-studying Lax's functional analysis, and I'm stuck in the chapter introducing spectral theory for unbounded self-adjoint operators. In his book, Lax proved the spectral theorem of this ...
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square root of positive elements preserve order

Let $A$ be a $C^*$ algebra. Show that if $0 \le a \le b$ then $\sqrt a \le \sqrt b$. I've shown that this is true in case $b$ is invertible, here is my proof: $$\|a^{1/2}b^{-1/2}\|^2 = ...
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Path - Geometry [closed]

I am currently completing the end of a Bachelor degree in pure mathematics. I would like to work on an interesting project (by myself) this summer in the field of spectral geometry. Does someone could ...
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35 views

Spectral Measures: Poisson

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}H\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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Pseudo-resolvent function

Let $\emptyset \neq D$ a open set in $\mathbb{C}$ and $J: D \to B(E)$ a continuos function such that $J(\lambda) - J(\mu) = (\mu - \lambda)J(\lambda) J(\mu)$ where $E$ is Banach space. We must show ...
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31 views

Compactness of the resolvent

I want to prove the following proposition: owning to the $ H^2_{loc}(\mathbb{R}^n)$ regularity when solving $\Delta^{(0)}_Vu=f\in L^2$, the compactness of the resolvent of $\Delta^{(0)}_V$ is a ...
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Bound self-adjoint operator

Assume we have a positive (so that we can take the square-root by functional calculus) self-adjoint operator $H: D(H) \subset \mathcal{H} \rightarrow \mathcal{H},$ then we can define ...
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Minimum of the Spectrum for a Closed Operator

If $T$ is closed is it true that if $\lambda=\min\{\sigma(T)\}$ and $((T-\lambda)f,f))=0$ then $(T-\lambda)f=0$? This is for a fixed $f\in\mathscr{H}$.
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Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: ...
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Show that $||(kI-T)^{-1}|| \le \frac{1}{d}$

Suppose that $T \in BL(H)$ where $H$ is a Hilbert Space. Let$k \in \mathbb{C}$. Let $d=distance(k,W(T)) \gt 0$. $W(T)=\{\lambda \in \mathbb{C}: \lambda=<Tx,x>, ||x||=1, x \in H\}$. Show that ...