Tagged Questions

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

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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 ...
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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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. ...