Trying to understand Axler's proof that every linear operator has an eigenvalue

Question

The proof is pictured in its entirety here. So in general, I think I understand the point of the proof and some of the steps, but I seem to have a gap in my understanding. It's at the very end where he says "which means that T - λjI is injective for at least one j" that I get confused. How is it that we know it's injective, and why exactly does that imply there must be an eigenvalue?

Is not injective for for a least one $j$? – Jonathan Davidson Jul 03 '17 at 06:00 — Jonathan Davidson, Jul 03 '17 at 06:00

Robert Lewis · Accepted Answer · 2017-07-03T07:31:05.827

5

Read your linked page a little more carefully. Axler says, "$T - \lambda_j I$ is not injective for at least one $j$. . . .

The product of injective operators is injective. Thus if

$\prod_1^m (T - \lambda_j I) \tag{1}$

is not injective, at least one of the factors fails to be injective as well.

edited Jul 03 '17 at 07:31

answered Jul 03 '17 at 05:51

Robert Lewis

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Trying to understand Axler's proof that every linear operator has an eigenvalue

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