Hi,
Where I can find (in which book) proof of the following theorem:
"The spectrum of a function in the algebra is the set of values of the function"
or how can I proof it?
Thanks for any help.
If C(S) is the Banach algebra of continuous functions on a compact set S, and $\displaystyle f\in C(S)$, the inverse of $\displaystyle f-\lambda1$ will be the function g given by g(t) = $\displaystyle \frac1{f(t)-\lambda}$ (provided that such a function exists). If f(t) is never equal to $\displaystyle \lambda$ then g will be defined on all of S, bounded (because S is compact) and continuous. Therefore $\displaystyle f-\lambda1$ has an inverse in C(S) and hence $\displaystyle \lambda\notin\sigma(f)$. But if there is a value of t for which $\displaystyle f(t)=\lambda$ then g is not defined at that value of t. In that case $\displaystyle f-\lambda1$ has no inverse in C(S) and hence $\displaystyle \lambda\in\sigma(f)$.