# Tagged Questions

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

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### Normal Operators: Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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### What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
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### Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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### Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
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### Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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### Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
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### An approximate eigenvalue for $T \in B(X)$.

This is a problem from Conwayâ€™s Functional Analysis: Definition An approximate eigenvalue for $T \in B(X)$ is a scalar $\lambda$ such that there is a sequence of unit vectors $x_{n} \in X$ ...
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### Spectral radius inequality

Suppose $A,B \in M(n \times n, \mathbb{C})$ or $A,B \in M(n \times n, \mathbb{R})$. Under wich hypothesis can I state that: $\rho(AB) \leq \rho(A)\rho(B)$ ?
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### Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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### How to find eigenfunctions of a linear operator (follow-up question)

This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ...
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### Selfadjoint Operator: Basic Criterion

For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way?
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### Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...