# Math Help - Spectrum of linear operators isn't empty

1. ## Spectrum of linear operators isn't empty

Hello,

i found this theorem in a book, but i can't understand the proof given there:

The thm. says, every linear Operator on a Banach-space: T:E->E has nonempty spectrum.

The proof is very short:
If T would have an empty spectrum, then the resolvent
$R_T : \mathbb{C}->L(E) , R_T (z)= (T-z*I)^{-1}$ is entire.

Since the Resolvent is bounded, and $\|\| T_T (z) \|\| ->0$ for $z->\infty$
by the thm. of Liouville, the resolvent is constant.

My Question is, why the Thm. of Liouville works in this case?
I know that this thm. is correct for functions $f: \mathbb{C}->\mathbb{C}$.
But in this case our function isn't complex valued.

Can someone explain it for me?

Regards

2. ## Re: Spectrum of linear operators isn't empty

It should be clear that if $f:\mathbb{C} \rightarrow X$ is holomorphic then for any $l\in X^*$ we have $lf:\mathbb{C} \rightarrow \mathbb{C}$ is holomorphic, so if $f$ is bounded, and since $l$ is bounded iff it sends bounded sets to bounded sets, then $lf$ is bounded so Liouville's theorem applies and gives $lf$ constant for all $l$. Assume $f$ is not constant ie. there exist $x\neq y$ such that $f(x)\neq f(y)$ then there is a functional that assigns $f(x)$ to a non-zero complex number (Hahn-Banach), so if $f(x),f(y)$ are linearly dependent we're finished, if not apply Hahn-Banach again assaigning $f(y)$ any other non-zero complex number. Either way we contradict the fact that $lf$ is constant.