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Why is the spectrum of a shift operator the closed unit disk?

Mathematics Asked on December 10, 2021

Consider the following text from Murphy’s: "$C^*$-algebras and operator theory":

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In example 2.3.2, why is $sigma(u) = Bbb{D}$ (= the closed unit disk)?

I can see that $sigma(u) subseteq Bbb{D}$ and $sigma(u^*) = Bbb{D}.$

Thanks in advance!

One Answer

$$ lambda in sigma(u) iff overline{lambda} in sigma( u^*).$$

Answered by Fred on December 10, 2021

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