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I'm supposed to find the point spectrum of an operator on the Banach space C(0;1) with a max norm. The operator is as follows:

T(f) = $\int_{0}^{x}f(t) \text{d}t-f(0)$.

I know that the point spectrum should be all non zero functions on the Banach space for which $\text{T}f=\lambda f$ is true, with $\lambda \in \mathbb{C}$. I have divided it into $\lambda =0$, for which I found no non zero functions and lambda differs from ${0}$, where I've calculated the differential equation with general solution

$f(x)=C\cdot e^{x/\lambda }$.

I don't know how to determine C and how to determine if the point spectrum is empty or not. Help would be greatly appreciated, thanks.

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The point spectrum is $\{-1\}$. Consider the equation $\displaystyle\int\limits_0^xf(t)dt-f(0)=\lambda f(x)$. If $\lambda=0$, we have $\displaystyle\int\limits_0^xf(t)dt=f(0)$, i.e. $f=0$.

If $\lambda\ne0$, then, as you write, by differentiating, we have $f(x)=ce^\frac{x}{\lambda}$. If we substitute in the original equation $x=0$, we get $f(0)(\lambda+1)=0$. If $\lambda\ne-1$, then $f(0)=0$, therefore $c=0$ and $f=0$. If $\lambda=-1$, then $f(0)$ may not be zero, and $f(x)=f(0)e^{-x}$.

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