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Operator whose spectrum is given compact set
Can spectrum “specify” an operator?

Prove that for each nonempty $M$ - compact subset of $\mathbf{C}$ exists operator $A:l_2 \rightarrow l_2$, such that $\sigma(A) = M$, where $\sigma$ denotes spectrum.

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    You want to assume $M$ to be non-empty since the spectrum of a (bounded) operator is never empty. Davide's construction and the one using multiplication operators are both mentioned in the linked thread "Can spectrum "specify" an operator?" (that question was the motivation for the question Damian linked to).2011-11-19

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