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

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Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
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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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23 views

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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Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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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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148 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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38 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 ...
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39 views

The distribution solution to $L^{+} u=0$ is u=0, where $L^{+}=-\frac{d}{dx}+x$?

Consider the creation operator $L^{+}=-\frac{d}{dx}+x$. If $u\in L^2(\mathbb{R})$ and is a distribution solution to the equation $L^{+}u=0$, then for any $\phi\in C_0^{\infty}(\mathbb{R})$ we have ...
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Spectral Representation of a Linear Combination of Sinoids

Let $\left\{X_t\right\}$ be the process defined by $$ X_t=\sum_{j=1}^nA(\lambda_j)e^{it\lambda_j} $$ in which $-\pi<\lambda_1<\lambda_2<\ldots <\lambda_n=\pi$ and ...
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Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its ...
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43 views

The Laplacian as the difference between mean value and a point

I am reading a book (Spectral Theory in Riemannian Geometry), which opens with an interpretation of the Laplacian. To get a better feeling about the Laplacian operator, consider for example a ...
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27 views

To prove this associative statement about spectral theory

Show that $\sigma(AB) \cup \{0\} = \sigma(BA) \cup \{0\}$ in general, and that $\sigma(AB) = \sigma(BA)$ if $A$ is bijective. I studied the associative statement of this somewhere but it did not ...
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39 views

Convert partial fraction to continued fraction?

Lets say you have a partial fraction of the form: $$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$ Can anyone explain to me, in mildly plain English, how to convert this partial ...
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39 views

Dirichlet eigenfunction cannot be extended to a continuous function on the closure

I need to show that there exist a bounded domain $ \Omega \subset \mathbb{R}^2 $, and a Dirichlet eigenfunction $u$ on $ \Omega$ such that u cannot be extended to a continuous function on $ ...
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29 views

Calabi's theorem

I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. ...
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Example of an operator whose spectrum satisfies this condition

Give an example of a Hilbert space $\mathcal{H}$ and an operator $A: \mathcal{H} \to \mathcal{H}$ satisfying $$\sigma \left( A \right) = \varnothing \neq \sigma \left( A^2 \right),$$ where $\sigma ...
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Finding the spectral representation of the Delta function given the Green function of an operator

I'm working through the problems of a book on Sturm-Liouville problems. In a problem I found the Green function for the following SLP2 problem $$\frac{-d^2g}{dx^2}-\lambda g=\delta(x-\xi)$$ ...
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45 views

Complete orthonormal system of eigenfunctions for trace-class nonnegative operator on a Hilbert space

In Da Prato/Zabczyk's book "Stochastic equations in inifinite dimension" I stumbled over the following paragraph: Let $Q$ be a trace class nonnegative operator on a Hilbert space $U$. [...] Note that ...
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33 views

Stability of least isolated eigenvalue under positive perturbation

This would be a useful theorem. Have you seen it anywhere? $\mathbf{Theorem:}$ Suppose a self-adjoint operator $H_0$ on a Hilbert space has a simple isolated least eigenvalue $0$ with separation ...
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Eigenvalues of a certain product of matrices with special structure

Sorry for cross-posting from MO. Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the ...
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Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,0]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,0] $. Using ...
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rad(T)=||T|| for non-normal T

It is well-known that for normal bounded operators $T$ on a Hilbert space one has $\mathrm{rad}(T)=\|T\|$ (where rad is the spectral radius). Are there any sufficient conditions under which a ...
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proving a quadratic form is closed

I'm trying to show that, given a spectral measure $d\mu_\psi(\lambda)$ for a self-adjont operator $A$, for the following quadratic form $$q_\lambda(\psi)=\int_{\mathbb ...
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operator ideals and their relationship to the geometry of Banach spaces, and other questions [closed]

Albrecht Pietsch wrote an excellent book on operator ideals in 1978, which I use frequently. But, I was reading the preface to his book, and I do not understand it. He writes (in English ...
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Proposed proof Decomposition theorem

Let $\mathcal{A}$ be a unital Banach algebra. I want to prove that if $a \in \mathcal{A}$ and spectrum $\sigma(a) = \sigma_{1} \cup \sigma_{2}$ where $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
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Is $\int |x\rangle \langle x|dx$ Mathematical?

I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ...
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36 views

What defines a Pseudospectral Method?

I'm trying to understand pseudospectral methods in the context of solving PDEs. However, I can't seem to find a solid definition for this. Is it simply a general term for solving a problem in parts: ...
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Simple example of operator spectral analysis.

I'm studying on Rudin - Functional Analysis the part related to Banach algebra Bounded/Unbounded operator. Specifically i've studied the part of Banach algebra (where the concept of spectrum is ...
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Spectral Decomposition of A and B.

I was given the following question in my linear algebra course. Let $A$ be a symmetric matrix, $c >0$, and $B=cA$, find the relationship between the spectral decompositions of $A$ and $B$. ...
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Spectral theorem for momentum operator

I'm trying to apply the spectral theorem to the explicit example of the momentum operator: \begin{equation} p:= i\frac{d}{dx} \end{equation} on the domain $D=\{\psi\in H^1(0,2\pi)\ |\ ...
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Looking for a general approach to proving compactness of linear operators.

Preparing for my exam in functional analysis, I often have to prove that certain explicitly given operators are compact. I now have a decent amount of operators of which I can prove the compactness, ...
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56 views

Some claims about spectra

I see the following equalities used sometimes, but couldn't find proofs. How is it done? $$\overline{Ran(L-\lambda\mathbb{1})}^{\perp}=ker(L^*-\overline{\lambda}\mathbb{1})$$ and ...
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Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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$C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
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Do you need to know a lot of (regular?) graph theory to get into spectral graph theory?

Do you need to know a lot of (regular?) graph theory to get into spectral graph theory? What are the prerequisites?
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Inverse of a dissipative operator

In the spectral theory of linear operators it is often helpful in the development of proofs for many results to proceed with the inverse of a (boundedly invertible) dissipative operator. For example, ...
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The subgroup of $GL_{n}(\mathbb{R})$ inside $B(1, 1)$ is $\lbrace I \rbrace$

Let $G \subset GL_{n}(\mathbb{R})$ be a subgroup such that $G \subset B$ where $$B = \lbrace M \in \mathcal{M}_{n}(\mathbb{R}), ||M-I|| < 1\rbrace $$ Let $g \in G$. I'm asked to show: The ...
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Show that $\lim \frac{f_\epsilon -1}{\epsilon}$ is purely imaginary if each $|f_\epsilon| = 1$

For $0< \epsilon < 1$, suppose we have complex numbers $f_{\epsilon}$ such that each $|f_{\epsilon}| =1$ and $$ \lim_{\epsilon \to 0} \frac{f_\epsilon - 1}{\epsilon} := a $$ exists. Prove ...
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Using Fubini's Theorem in Contour Integrals proof

I have a few questions regarding the following proof: Suppose that $\mathcal{A}$ is a unital Banach algebra, and that $g$ is a complex-valued function which is analytic on $\sigma(a)$ while $f$ is a ...
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Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
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1answer
54 views

Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
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Is this possible to decompose 2 arbitrary unitary matrices as: $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$?

I guess this statement is true, but I can't prove it: "For any 2 arbitrary unitary matrices $U_{1}$ and $U_{2}$, we can always decompose them into $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$, where $A$ ...
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Difference between an eigenvalue and a spectral value

What is the difference in the definition of a spectral value and an eigenvalue. My notes from functional analysis says $\lambda$ is an eigenvalue of an operator $A$ if $\,\exists \, x \in ...
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Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a function: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad ...
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46 views

Spectrum of compact operators on an infinite dimensional normed space

The question is as follows: Let $T:X \to X$ be a compact linear operator on a normed space. If the $dim X= \infty$ then show that $0 \in \sigma(T)$. My attempt: Suppose on the contrary that $0 ...
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Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
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About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...