# Tagged Questions

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### Generalized eigenspaces of a compact operator are finite dimensional

Let $T : H\rightarrow H$ be a compact operator on a Hilbert space $H$. Say that $\lambda \in \mathbb C$ is a generalized eigenvalue of $T$ if there is some $n \geq 1$ such that $(\lambda - T)^n$ is ...
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### Property of sequence of eigenvalues of an operator.

For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
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### norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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### Example of a normal operator which has no eigenvalues

Is there a normal operator which has no eigenvalues? If your answer is yes, give an example. Thanks.
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### Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
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### Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$\mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy$$ so that it has all its ...
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### What are the Eigenvectors of the curl operator?

The curl operator $\vec\nabla\times\mathbb{1}$ can be written as a skew-symmetric 3x3 matrix \mathrm{curl} = \begin{pmatrix}0 & -\partial_z & \partial_y \\ \partial_z & 0 & ...
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### Has this operator $0$ as an eigenvalue / where is my error?

I know of a theorem that tells me, that every compact linear operator on an infinitedimensional Hilbert space has to have the eigenvalue $0$. On the other hand I have the operator \begin{eqnarray*} ...
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### eigenvalue check

I have a question regarding operator theory and would be glad if someone could help. I have a linear operator $A$ that is non-self-adjoint, unbounded and is densely defined in a Hilbert space $H$. I ...
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### meaning of this operator ??

given the operator $P_{\Lambda } = (f\in L^{2}(R)^{even}| f(q)=0 , |q| \ge \Lambda)$ what does it mean? the operator $P_{\Lambda}$ acts over a function $f(q)$ by setting this (even) function to ...
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### Find the minimum value of the maximum eigenvalue of operator A?

So we are given the following: Operator $A$ with $Au=-u''$; $u \in D_A = \{u\colon[a,b]\rightarrow R,u\in C^2([a,b]),u(a)=u(b)=0\}$; $D_A$ is dense in $L^2((a,b))$. Find the minimum value that is ...
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### Eigenvalues of compact operators and his adjoint.

Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$. Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...
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### Sturm-Liouville Theorem

I was reading the Wiki page on the Sturm-Liouville theory. Why are those tenets true? Are there any (not too advanced) reference material? I have also read that "There are countably infinite ...
I'm trying to solve the following exercise: Let $f\in\mathcal{C}([0,1])$ and let $T$ an operator such that $Tf(x)=\int_0^1(x-t)f(t)dt$. I have proved that $T$ is a bounded linear operator and, by ...