# Tagged Questions

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

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### Expansion of an eigenfunction $\psi$ into a Fourier series

I was going through a paper by Jean Bourgain and it states that an eigenfunction $\psi$ of the Laplacian $\Delta$ with eigenvalue $-\lambda$ on the flat $n-$torus ...
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### Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using, H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term ...
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### irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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### Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when ...
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### Maximal Accretive Implies Injective

I'm having trouble proving that $B$ is essentielly maximal accretive implies that there existes $a>0$ such that $(A^∗+a Id)$ is injective. where B is the closure of the operator A and A* is ...
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### Equality of two measures

I am trying to prove the following statement. Let $F$ and $G$ be two finite measures on $((-\pi,\pi],\mathcal{B}((-\pi,\pi]))$ such that ...
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### Fokker Planck operator is accretive

I would like to show that the Fokker Planck operator with domain $C_0^\infty \left(R^{2n}\right)$ defined by $$K= v.∂_x − (∂_xV (x)).∂_v + (−∂_v +v/2).(∂_v +v/2)$$ is essentially maximal accretive ...