Let $A\colon E\to E$ definied by $A(f)(x)= \int\limits_0^x f(t) dt$. I have to find the spectrum of $A$ in the cases $E=C[0,1]$ and $E=L_2[0,1]$. I have proved that $A$ has no eigenvalues, but I can't find full spectrum.
Spectrum of $\int\limits_0^x f(t) dt$ operator
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functional-analysis
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1Right. What space are you working on? – 2011-11-21
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0Sorry. I need to find spectrum in both $C[0,1]$ and $L_2[0,1]$. – 2011-11-21
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6You can find an inverse of $A-a I$ by solving $g(x) = (\int^x f) - a f(x)$: differentiating gives an ode that you can solve explicitly, and then find an expression for $f$ even when the functions are not differentiable. – 2011-11-21