All Questions
Tagged with operator-theory eigenvalues-eigenvectors
228
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Spectral Theorem applied to Laplace-Beltrami Operator
Consider $(M,g)$ a compact Riemannian manifold, and $L^2(M)$ the Hilbert space of the square-integrable functions with respect to the Riemannian volume form. I want to study the eigenvalue problem for ...
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Conditions for the existence of eigenvectors for a positive operator on a Banach Space
Suppose $X$ is a Banach space and $K \subset X$ is a normal cone with interior. Suppose further that $A$ is a non-compact strongly positive operator, $A(K) \subset$ int$K$. It is well known that under ...
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Isolated Eigenvalues of Finite Multiplicity in the Spectrum of a Self-Adjoint Operator with Compact Perturbation
I am studying spectral properties of operators, specifically working through Ronald G. Douglas' "Banach Algebra Techniques in Operator Theory." While doing exercise 5.15, I encountered the ...
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What are the spaces whose linear operators admit eigen"vectors"?
I know that linear operators over $n$-dimensional, $\mathbb{R}-$vector spaces admit $n$ eigenvalues (in the algebraic closure of $\mathbb{R}, \mathbb{C}$) and eigenvectors. The proof of this is just ...
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Reconstruction of an operator given the eigenfunctions and eigenvalues
I am interested in operator theory, in particular if I know the sequence of eigenvalues $\{\lambda_n\}_{n=1}^\infty \subset \mathbb{R}$ and eigenfunctions $f_n \subset X$ of a self-adjoint ...
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Applying Spectral Mapping Theorem to determine if $f(T)$ is compact
Given a self-adjoint, compact operator $T$ and a continuous $f:\sigma(T)\to\mathbb{R}$, I'm trying to determine the conditions under which $f(T)$ is also compact. I know of the Spectral Mapping ...
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What is the importance of the largest eigenvalue / spectral radius of a symmetric positive matrix being equal to 1? Particularly in attention.
It is often said, that if the spectral radius of a matrix $\boldsymbol{A}$ is equal to $1$, the matrix has "regularizing" properties for the matrix product $\boldsymbol{Ax}$ for a vector $\...
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
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Computing Point Spectrum and Resolvent of Multiplication Operator on [0,1]
Not Homework - Just Personal Study
I have (hopefully) solved the problem I am posting, I would like to know if my proof is correct, and if not, what mistake did I make.
Consider $C[0,1]$ with the ...
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Proof of the convergence of the Rayleigh-Ritz Method
In this article The convergence of the Rayleigh-Ritz Method in quantum chemistry by Bruno Klahn & Werner A. Bingel they have at page 11
Let $H_B$ be that Hilbert space which can be obtained as the ...
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Why Spurious Eigenvalues comes from mixed states?
In this book Many-Electron Approaches in Physics, Chemistry and Mathematics they have the following definition at page 36
Definition 1 (Spurious spectrum) A real number $\lambda \in(-1,1)$ is called ...
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Eigenvectors of commuting operators on finite complex hilbert spaces.
This topic has been discussed in various other posts but I haven't been able to find specifically what I'm looking for. I've seen the proof in the case of an infinite number of commuting operators, ...
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every uniformly continuous semigroup of linear operators on finite dimensional space does have a vector with unbounded orbit
Assume that $\mathcal{T}=\{T(t)\}_{t\geq 0}$ be uniformly continuous semigroup of bounded linear operators . It is known that there is a bounded linear operator $A$ such that $T(t)=e^{tA}$. Also
the ...
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Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.
Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
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How is a spectral subspace of a bounded linear operator defined?
Let $A$ be a bounded linear operator on some Hilbert space. In a previous question (How to interpret spectral projections?) I learned that the spectral projectors, which are defined using the Borel ...
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