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Could anyone help me with the following exercise? Any help will be very welcome:

Given a sequence $\{\alpha_n\}$ so that $\alpha_n \to 0$, show that exists a compact operator $T$ with spectrum $\sigma(T)=\{\alpha_n\} \cup \{0\}$

[Edit 1]: I found a useful construction in this answer

[Edit 2]: $T:l^2 \rightarrow l^2, T((a_n)_{n=1}^\infty)=(\lambda_na_n)_{n=1}^\infty$ does the job!

tomate
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jun 28 '23 at 22:43
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    Hint: Try a diagonal operator on $\ell^2(\Bbb N)$. – s.harp Jun 29 '23 at 07:05

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