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

$\displaystyle R_T : \mathbb{C}->L(E) , R_T (z)= (T-z*I)^{-1}$ is entire.

Since the Resolvent is bounded, and $\displaystyle \|\| T_T (z) \|\| ->0$ for$\displaystyle z->\infty$

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 $\displaystyle f: \mathbb{C}->\mathbb{C}$.

But in this case our function isn't complex valued.

Can someone explain it for me?

Regards