Spectrum of an element in the disk algebra.

Question

Consider the disk algebra $A$ of continuous function on the unit disk $D$ that are analytic on the interior of the disk.

Is it true that $\sigma_A(f) = f(D)$ for $f \in A$? A simple yes or no suffices. I think that the same proof of the theorem that says that $$\sigma_{C(X)}(f) = f(X)$$ for $X$ compact Hausdorff modifies but I want a quick sanity check. Thanks!

Answer 1

Yes, this is correct. It follows from the fact that a function in $A$ is invertible if and only if it doesn't equal to zero at any point in $D$.