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Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function $\lambda \mapsto R(\lambda,A)$ and the Banach version of Liouville's theorem.

I am just asking if there's another approach to prove that the spectrum of $A$ is not empty without using Liouville's theorem.

This will make me better understand the power and beauty inside Liouville's theorem and complex analysis in general.

user165633
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    How many proofs do you know for the Fundamental Theorem of Algebra? Louiville's Theorem supplies the best proof for the Fundamental Theorem that I know about. And this result of non-empty spectrum goes further. Here's a fascinating stackexchange link concerning finding an algebraic proof of the Fundamental Theorem of Algebra. Not a simple matter ... http://math.stackexchange.com/questions/165996/is-there-a-purely-algebraic-proof-of-the-fundamental-theorem-of-algebra – Disintegrating By Parts Jun 19 '14 at 18:01
  • @T.A.E. Thanks, that was a great link. This maybe just teaches us that it is not good to isolate areas in mathematics. I think J. Hadamard said "The shortest path between two truths in the real domain passes through the complex domain". I think physicists also can't work without complex analysis, because quantum physics can't be described without complex numbers. – user165633 Jun 19 '14 at 21:33

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