Find spectrum of the operator and an explicit form for the solution

Asked May 03 '13 at 00:40

Active May 03 '13 at 00:40

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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) Suppose $\lambda^{-1} \not\in \sigma(T)$ and $g \in L^2[a,b]$. Obtain an explicit form for the solution of $u=g+\lambda Au$ and write it in the form $u(x)=\phi(x)+\int^{2\pi}_{0}\Gamma(\lambda,x,t)\phi(t)dt$

asked May 03 '13 at 00:40

pyq234

So, What is the definition of point spectrum? Basically, You need to solve the integral equation $ Au=\lambda u $. – Mhenni Benghorbal May 03 '13 at 01:17
See here for the technique. – Mhenni Benghorbal May 03 '13 at 01:32

Find spectrum of the operator and an explicit form for the solution

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