# Tagged Questions

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Given a bounded set $\mathcal A\subset \Bbb R^{n x n}$. The joint spectral radius is given by: $\sigma(\mathcal A)$=$limsup_{m\to\infty}(sup_{A\in\mathcal A^m} \rho(A))$ where $\rho$ is the normal ...
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### Spectrum of an element

I'm having a little trouble calculating the spectrum of an element: specifically, the element $f(x) = \frac{1}{x}$, as an element of the bounded continuous functions from $[1, \infty)$ with pointwise ...
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### Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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### Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
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### Eigenvalues of a connected graph $G$ are greater than or equal to $-1$ iff $G$ is perfect?

Consider $P_G$ as the characteristic polynomial of the adjacency matrix of the connected graph $G$. It is easy to prove that $P_{K_n}(x)=(x-n+1)(x+1)^{n-1}$, so all of the eigenvalues of a perfect ...
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I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem? The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ... 1answer 50 views ### Well definededness of integration with respect to a projection valued measure Let$(X,\mathcal{F})$be a measurable space and let$E:\mathcal{F}\to\mathscr{B(H)}$be a spectral measure. Let$\phi\in B(X)$be a simple function whose image is ... 3answers 335 views ### Nonexistence of a cyclic vector for a representation on$\ell^2(\mathbb{Z})$Let$S$be the bilateral shift on$\ell^2(\mathbb{Z})$and let$T = S + S^*$. I want to show that there is no cyclic vector for the representation of$T$on$\ell^2(\mathbb{Z})$i.e.$\forall x\in ...
Let $\mathcal{H}$ be a Hilbert space and let {$e_j$}$_{j\in \mathbb{Z}}$ be an orthonormal basis for $\mathcal{H}$. Define a linear operator $T$ on $\mathcal{H}$ by $T(e_0) = 0$ and $T(e_j) = e_{j+1}$ ...
I have the following problem: Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...