Gelfand's Formula. $r(T)=\lim_{n \to\infty}\sqrt[n]{\|T^{n}\|}$

Question

Can you indicate me a material where I cand find the proof of Gelfand's Formula. I heard that there is a proof with polynomials.

Gelfand's Formula :

If $T \in B(X)$ then: $$r(T)=\lim_{n \to\infty}\sqrt[n]{\|T^{n}\|}$$

thanks for your help :-)

Answer 1

I do not know of any proof using just polynomials. All the proofs I have seen involve complex analysis to a certain extent, either Laurent series expansions, Liouville's theorem, or Hadamard's Theorem.

I am pretty sure that most books on functional analysis which cover Banach spaces and/or Banach algebras will contain a proof, for example:

G. F. Simmons - Introduction to Topology and Modern Analysis - page 312 in my (old) edition.

Riesz and Nagy - Functional Analysis - section 149 (page 425 in my Dover edition).