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Find the spectrum of the operator $$ \begin{split} A & \colon C[0,1] \rightarrow C[0,1] \\ & f \mapsto (Af)(x) := f(x) + \int_0^x f(t)dt \end{split} $$

P.S.: I know the spectrum of $A_1\colon f \mapsto \int_0^x f(t)dt$

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The spectrum comes from analyzing $A-\lambda I$. Since $A = A_1+I$, you can see that $A-\lambda I = A_1-(\lambda-1)I$. It follows that $\sigma(A) = \sigma(A_1) +\{1\}$.

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$\sigma(A) = \lbrace 0 \rbrace + \lbrace 1 \rbrace = \lbrace 1\rbrace? $ – daisy Dec 18 '12 at 18:01
Yes. A fairly standard notation for sets $A,B \subset \mathbb{R}$ is $A\square B = \{a \square b| a\in A, b \in B \}$, where $\square$ is some binary operator such as $+$. – copper.hat Dec 18 '12 at 18:10

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