## Month: January, 2014

### K-spectral sets and the holomorphic functional calculus

In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.

In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).

### Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Read the rest of this entry »

### 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,

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