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Consider $K : C^0[0, 1] → C^0[0, 1]$ given by $K(f)(x) = \int_{0}^{x}f(t) dt.$

Check it is well-defined, linear and continuous.

Find $||K||, K(C^0[0, 1]), σ_p(K), σ_c(K)$ and $σ_r(K)$.

Also, check the spectral radius formula, namely, $\rho(K) = lim_{n\to \infty} ||K^n||^{1/n}$

For the last exercise, I found this solution Spectral radius of the Volterra operator, and a I think that it is ok. And I think that $||K||=1$ by the length of the interval. I don't know how I can solve the others. Can you help me please?

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Yoooopi
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Hints: for the range, note that a function $f$ on a closed, bounded interval $[a, b]$ is absolutely continuous on $[a, b]$ if and only if it is an indefinite integral over $[a, b].$ For the kernel, use the FTC, which applies to say that since $f$ is continuous on $[0,1],\ F(x)=\int^x_0f(t)dt$ then $F'(x)=f(x).$

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