### 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).