Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$

Question

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.

Answer 1

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.