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

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51 views

Find the eigenvalues of the operator T.

I have the following problem, "Suppose that $X=\ell^1$ and define the operator $T\in B(X)$ as follows: $$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, ...
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4answers
180 views

Left Shift Operator Spectrum

Consider the Hilbert space $\mathcal{H}=l^2(\mathbb{Z})$ and define the left shift operator $\mathcal{L}:\mathcal{H} \rightarrow \mathcal{H}$ by $$ \mathcal L (a_n) = (b_n) \qquad \text{ where } ...
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1answer
20 views

Let $f$ and $g$ linear operators where $f$ and $g$ commute and $f$ has simple spectrumm, then there is $P$ a polynomial such thah $g=P(f)$.

Let $f : \mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ be a linear operator with a simple spectrum, furthermore, let $g : \mathbb{C}^{n}\rightarrow \mathbb{C}^n $ be a linear operator such that $f$ and ...
2
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1answer
47 views

Estimate spectral radius of operator product

In my research problem, I have to estimate the spectral radius of the following operator $\chi A$ where $\chi$ is a scalar function taking values 0 or 1 and $A$ is an operator. I can compute ...
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1answer
21 views

Are these critical points minima to the variational problem?

Let $\Omega\equiv (0, 1)\times(0, 1)\subset\mathbb{R}^2$ and consider the variational integral \begin{equation*} I[u]\equiv\int_{\Omega}\frac{1}{2}|Du|^2\ \mathrm{d}x-\frac{5\pi^2}{2}|u|^2\ ...
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1answer
40 views

Topological characterization of the range of a bounded normal operator

Let $T$ be a bounded normal operator on a Hilbert space $H$. I want to prove the following statement: $\text{ran}(T)$ is closed if and only if 0 is not a limit point of $\sigma(T)$. I tried to use the ...
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1answer
14 views

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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1answer
42 views

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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0answers
12 views

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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0answers
28 views

Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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0answers
22 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 ...
3
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56 views

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 ...
2
votes
1answer
30 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 ...
2
votes
1answer
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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0answers
16 views

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 ...
0
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1answer
67 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
2
votes
2answers
99 views

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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1answer
42 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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2answers
25 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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1answer
35 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 ...
2
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0answers
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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37 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 $ ...
2
votes
2answers
23 views

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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0answers
30 views

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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26 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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0answers
19 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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0answers
15 views

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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7 views

Holonomy group at a base point of a locally symmetric space

What is the holonomy group at a fixed base point $x_{0} \in X$, where $X$ is a locally symmetric space? What is the holonomy conjugacy class associated to a geodesic in $X$? I was trying ...
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1answer
16 views

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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63 views

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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32 views

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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166 views

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 ...
4
votes
1answer
77 views

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 ...
2
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0answers
39 views

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 $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
3
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2answers
29 views

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 $B$. ...
0
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1answer
32 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: ...
3
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2answers
54 views

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 ...
2
votes
1answer
44 views

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)\ |\ ...
1
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1answer
125 views

Proof of Spectral Theorem

There are a number of results called Spectral Theorems. This question deals with the Linear Algebra result on normal operators, which has the self-adjoint case as a particular case. In class, we saw ...
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1answer
39 views

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, ...
0
votes
1answer
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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36 views

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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0answers
52 views

$C^k$ 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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1answer
30 views

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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28 views

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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2answers
79 views

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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votes
1answer
31 views

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 ...
3
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1answer
68 views

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 ...
2
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1answer
43 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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32 views

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. ...