# 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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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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The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: \... 2answers 929 views ### Spectrum of shift-operator Hoi, consider the Hilbertspace l^2 and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ... 1answer 191 views ### Min-Max Principle \lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \} - Explanations In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ... 0answers 47 views ### Linearization of PDE: 0 is an eigenvalue since all translates of travelling waves are also travelling waves Consider the following PDE: u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$where f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ... 3answers 579 views ### Spectrum of a nilpotent operator Let X be a Banach space and A:X\to X be a bounded operator such that A^n=0 for some n\in \mathbb{N}. Is the spectrum of A finite, countable ? 1answer 2k views ### How to find eigenfunctions of a linear operator (follow-up question) This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ... 1answer 88 views ### Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ... 1answer 76 views ### Spectral Measures: Embedding This thread is just a note! Given a Hilbert space \mathcal{H}. Consider a normal operator:$$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$And its spectral measure:$$E:\mathcal{B}(\mathbb{C})\...
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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, ...