Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

learn more… | top users | synonyms (1)

1
vote
0answers
115 views

Inverse of bounded self adjoint operator on HS is self adjoint?

Let $A=A^{*}$ be a bounded self adjoint operator on a Hilbert space $\mathcal{H}$ with Range Ran$(A) = D$ dense in $\mathcal{H}$. $A$ is injective, since Ran$(B) \perp ker(B^{*}) = ker(B)$. So ...
1
vote
0answers
39 views

Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
1
vote
0answers
39 views

Properties of the operator $ (Tu)(n)=u(n+1)-u(n-1)$

Let $T:l^2(Z,R)\to l^2(Z,R)$ the linear bounded operator defined by: $$ (Tu)(n)=u(n+1)-u(n-1)$$ a)Prove that the image of T is dense. b)Prove that $\forall \lambda\neq 0\quad t-\lambda I$ is ...
1
vote
1answer
61 views

Using Weyl sequences to prove relation between quadratic form and spectral radius

I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$ holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula ...
1
vote
0answers
95 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
1
vote
0answers
42 views

Solving an inverse spectral problem

In order to solve the inverse spectral problem: $$ -y''(x)+q(x)y(x)= \lambda _{n}y(x) $$ If we want to obtain $ q(x) $ what we should need about the spectrum? a) The eigenvalue staircase $ ...
1
vote
0answers
91 views

Examples of deeper results in finite-dimensional vector spaces?

this one is a bit inverted! So I am busy doing an advanced undergrad course in Linear algebra, and it is going very well, the problems in the book seem fairly routine. To be able to see if I am any ...
1
vote
0answers
42 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
1
vote
0answers
15 views

Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
1
vote
1answer
241 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
1
vote
0answers
26 views

looking for “invertibility and singularity”

Dear *friends* Many monts ago,i searched a lot the book of Robin Harte "invertibility and singularity". this book contains a lot of demonstrations that i need in my master. it focus on banach ...
1
vote
0answers
49 views

Resolution of Identity

I am reading the twelfth chapter of Rudin's Functional Analysis and am unable to understand two comments he makes in passing in the section on Resolution of Identity. Let E be a resolution of identity ...
1
vote
0answers
33 views

What is twisted triangular two-torus also called a triangular doughnut?

In "A Geometry of Music" by Dmitri Tymoczko Oxford 2010, the authors says that mathematicians refer to a particular lattice in what mathematicians would call "the interior of a twisted triangular ...
1
vote
0answers
82 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
1
vote
0answers
131 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
1
vote
1answer
116 views

Spectral density function in stationary process

I have the following argument in my Time Series class notes: Let ${u_t}$ be a mean zero covariance stationary process. Define $\gamma(j) = \mathbb{E}u_tu_{t-j}$ and $Y_t = \mu +C(L)u_t$ where ...
1
vote
0answers
62 views

Determining the spectral density?

Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and ...
1
vote
0answers
45 views

Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
1
vote
1answer
55 views

Relationship between spectrum and Norm of bounded linear maps .

I am reading the following paper : http://www2.icmc.usp.br/~sma/cadernos/toc9.1/292.pdf In the second paragraph the author introduces a new operator $\|x\|_{T,\epsilon}$ , which i don't really ...
1
vote
0answers
69 views

Properties of the spectrum

Let $\rho$ denote the resolvent of a closed operator and if $\lambda \in \rho(A)$, define $R(\lambda,A) := (\lambda I -A)^{-1}$. If $\mu$ is sufficiently small ...
1
vote
1answer
41 views

What's wrong with this spectrum of a “scalar product” in $l^2$?

Let $T\in B(l^2)$ be s.t. $Tx=(\alpha_1 x_1, \alpha_2 x_2, \cdots )$, where the set of all $\alpha$ is dense in $[0,1]$. I've shown that the set of all eigenvalues is $A=(\alpha_j)_1^\infty$. The ...
1
vote
0answers
39 views

Normalized Cuts and Spectra

I'm looking for a fleshed out proof of the following theorem. Theorem: Let $G=(V,\mathbf{W})$ be an undirected, edge-weighted graph with normalized Laplacian $\mathbf{L}_N$. Furthermore, let ...
1
vote
1answer
102 views

Spectrum of a shift composed with a multiplication operator on a vector valued Banach space

Let us consider the space $L_2(\mathbb{R} \times [0,1]; \mathbb{R}^n)$, i.e functions taking values in $\mathbb{R}^n$ and in $L_2$ . Suppose $T$ is a bounded linear operator defined as follows: ...
1
vote
0answers
54 views

Compute Rayleigh quotient for ODE

I am trying to find Rayleigh quotient for this equation: $u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where $0 \le r \le 1$. Is there any way to compute eigenvalue ...
1
vote
0answers
55 views

Uniform Poincaré-Wirtinger constant for diffeomorphic domains?

Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
1
vote
1answer
227 views

Similar matrix proof

$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
1
vote
0answers
48 views

Strict positivity on the diagonal of a particular integral kernel: A question from Simon's Schrödinger Semigroups

This is a question pertaining to a (formerly?) open question from Barry Simon's Schrödinger Semigroups. In Theorem C.5.2 (page 504) of that publication, the existence of a specific function ...
1
vote
0answers
56 views

When will the equality of Neumann's Trace inequality holds?

I am now studying the Neumann's trace inequality. Some of the literature said that the equality holds when the two matrices have simultaneous ordered spectral decomposition. Yet they don't really ...
1
vote
0answers
239 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
1
vote
0answers
232 views

Orthogonal projection and normal operators

Let $G$ be normal operator with compact resolvent such that $\ker G$ is different from $\{0\}$. Now Let $P$ be the orthogonal projection onto $\ker G$ and consider $G' = G + P$. Please, I want an ...
1
vote
1answer
50 views

Well definededness of integration with respect to a projection valued measure

Let $(X,\mathcal{F})$ be a measurable space and let $E:\mathcal{F}\to\mathscr{B(H)}$ be a spectral measure. Let $\phi\in B(X)$ be a simple function whose image is ...
1
vote
0answers
116 views

relation between inner product and spectrum

There is a question that puzzles me, so may be someone here has an answer. Assume we have a symmetric operator $A$ that is defined on a space $D$ that is dense in $L^2$, so $A:D\rightarrow L^2$, and ...
1
vote
0answers
98 views

Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?

Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of ...
0
votes
2answers
31 views

Example: Algebraic Multiplicity vs Geometric Multiplicity

Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity?
0
votes
3answers
197 views

Invertibility of $I-A$ if the spectral radius of the operator $A$ is less than $1$

I want an explication of the following fact: If the spectral radius of a bounded operator $A$ on a Banach space is less than one, then $I - A$ is invertible.
0
votes
2answers
77 views

Trace of a diagonalized matrix

Why do I have: $Tr(SDS^{-1})=Tr(D)$?
0
votes
2answers
68 views

Symmetric Operator $\iff$ Real Spectrum

One the one hand not every symmetric operator has real spectrum: $$A\text{ symmetric}\nRightarrow\sigma(A)\text{ real}$$ (In fact this is true only for the self adjoint ones.) Does the converse fail ...
0
votes
2answers
44 views

Basic Criterion on Selfadjointness

How to prove the following in a neat subjective way: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$
0
votes
3answers
99 views

The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$

On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
0
votes
1answer
48 views

Spectrum of T in $B(\ell^2)$

Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows: $$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$ What is spectrum of T in $B(\ell^2)$?
0
votes
1answer
37 views

The spectrum of $C(K)$ where $K$ is a compact Hausdorff space

Let $K$ be a compact Hausdorff space, what's the spectrum of $f\in C(K)$? I don't know how to start.
0
votes
1answer
210 views

Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

in the Wu-sprung model, given a Hamiltonian in one dimension $$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$ we can define the function $ f(x) $ implicitly as $$ f^{-1}(x)= 2\sqrt{\pi} ...
0
votes
1answer
149 views

The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$? Edit: What if the matrix has positive ...
0
votes
1answer
278 views

Approximate Point Spectrum is subset of Spectrum

I'm trying to prove that if $\lambda$ an approximate eigenvalue of $T$ then $\lambda \in \sigma(T)$, but I can't work out how to do it. Could someone give me a hint, or point me in the direction of a ...
0
votes
1answer
35 views

Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
0
votes
1answer
61 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
0
votes
1answer
50 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
0
votes
1answer
45 views

How to find the spectrum $\sigma_p(P)$

How to find the spectrum $\sigma_p(P)$: Let $P:H\rightarrow H$ be an orthoprojection, $P\neq 0, P\neq I$. could you please help
0
votes
2answers
45 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
0
votes
1answer
37 views

Why is this spectrum the closure?

I've started to learn about spectral theory and I'm looking at some examples. The spectrum is defined as $$ \sigma(a) = \{\lambda \in \mathbb C | a-1\lambda \text{ is not invertible} \}$$ If $A= ...