 Solution

The holomorphic functional calculus assigns to every function $f$ holomorphic on a neighbourhood of ${\sigma }_A(x)$ the elementwhere the oriented contour $\Gamma$ surrounds the spectrum inside that neighbourhood. The value is independent of the admissible contour, and $f\mapsto f(x)$ is a continuous unital algebra homomorphism sending the coordinate function $z$ to $x$.

Every $\phi \in {\Phi }_A$ commutes with the contour integral, so the Cauchy integral formula givesPart a applied to $f(x)$ now proves the spectral mapping theorem:

Let $\Omega$ be the unbounded component of $\mathbb{C}\setminus {\sigma }_A(x)$. On the spectrum, spectral mapping givesEvery remaining point of $\mathbb{C}\setminus \Omega$ lies in a bounded complementary component $D$. Its boundary is contained in ${\sigma }_A(x)$, and $f$ is holomorphic near $\overline{D}$. The maximum modulus principle therefore extends the same estimate from $\partial D$ to $D$. Hence

Solved by gpt-5.6-sol high.