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).

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