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Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint.

Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let $D_o$ and $D_1$ be the disks in the complex plane of the same radius $\|a\|$ centred at $0$ and $1$, respectively. Then $\operatorname{Sp}a \subset D_o \cup D_1$.

Zev Chonoles
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rese
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    $Spa$ is the spectrum of $a$? And are the disks supposed to be closed or open? – Daniel Fischer Jul 06 '13 at 19:45
  • $Sp a$ means spectrum of a. And discs are closed. – rese Jul 06 '13 at 19:51
  • Just to make sure we're on the same page, the Banach algebra has a unit $e$, and the spectrum of $a$ is the set ${\lambda \in \mathbb{C}\colon \lambda\cdot e - a \text{ is not invertible}}$. Or are you referring to the spectrum of the operator $T_a \colon x \mapsto a\cdot x$? (Not that it makes much difference.) – Daniel Fischer Jul 06 '13 at 19:56
  • Use \operatorname{} to get the correct spacing and font. Just writing Spa means "the product of variables named $S$, $p$, and $a$", whereas \operatorname{Sp}a means "an operator named $\operatorname{Sp}$ applied to $a$" – Zev Chonoles Jul 06 '13 at 19:57
  • @Daniel Fischer: $\operatorname{Sp}a = {\lambda \in \mathbb{C}\colon \lambda\cdot e - a \text{ is not invertible}}$ – rese Jul 06 '13 at 20:19

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This is Theorem 3.2.3 in "Fundamentals of the Theory of Operator Algebras" by Kadison and Ringrose. Their proof is instructive and I would suggest trying to look at it:

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Zev Chonoles
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pre-kidney
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  • Thanks for adding the image; I hope it is "fair use" :) – pre-kidney Jul 06 '13 at 20:25
  • I believe that it is - from the US copyright office: "The 1961 Report of the Register of Copyrights on the General Revision of the U.S. Copyright Law cites examples of activities that courts have regarded as fair use: 'quotation of excerpts in a review or criticism for purposes of illustration or comment; quotation of short passages in a scholarly or technical work, for illustration or clarification of the author’s observations...'" – Zev Chonoles Jul 06 '13 at 20:36