### Advanced Analysis, Notes 18: The holomorphic functional calculus I (motivation, definition, line integrals of holomorphic Banach-space valued functions)

This course, Advanced Analysis, contains some lectures which I have not written up as posts. For the topic of Banach algebras and C*-algebras the lectures I give in class follow pretty closely Arveson’s presentation from “A Short Course in Spectral Theory” (except that we do more examples in class). But there is one topic  – the holomorphic functional calculus -for which I decided to take a slightly different route, and for the students’ reference I am writing up my point of view.

Throughout this lecture we fix a unital Banach algebra $A$. By “unital Banach algebra” we mean that $A$ is a Banach algebra with normalised unit $1_A$.  For a complex number $t \in \mathbb{C}$ we write $t$ for $t \cdot 1_A$; in particular $1 = 1_A$.  The spectrum $\sigma(a)$ of an element $a \in A$ is the set

$\sigma(a) = \{t \in \mathbb{C} : a- t \textrm{ is not invertible in } A\}.$

The resolvent set of $a$, $\rho(a)$, is defined to be the complement of the spectrum,

$\rho(a) = \mathbb{C} \setminus \sigma(a)$.