Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in \text{spec}\,a} |z| = \limsup_{n \to \infty} \text{N}(a^n)^{1\over{n}}$$Much thanks.
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$\begingroup$ I don't understand these terms and how they relate to the title. $\endgroup$– Will SawinAug 30, 2015 at 17:01
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5$\begingroup$ This is just the spectral radius calculation for unital Banach algebras. I suggest you look at any number of standard textbooks or googling "spectral radius proof banach algebra". $\endgroup$– Chris RamseyAug 30, 2015 at 17:32
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The limit always exists and this is called Gelfand's formula for the spectral radius. Any textbook on spectral theory has a proof of it, for example N. Burbaki, Theories spectrales. This one is also very good: B. Aupetit, A primer on spectral theory, Springer-Verlag, New York, 1991. For more detail about such formulas you may look at this paper, for example: Spectral inclusion and analytic continuation, Bull. London Math. Soc., 31 (1999), 722-728.
answered Aug 30, 2015 at 20:36