C-spectral sets and related estimates

This dissertation deals with the task of finding norm bounds for functional calculus maps. More precisely, given an element A of a unital Banach algebra and an appropriate numerical function f, we try to understand how the norm of the element f(A) is controlled by the supremum norm of f on certain regions in the complex plane, a notion closely connected to c-spectral sets. In particular, motivated by Crouzeix’s conjecture (2004), we are interested in the case where the region is given by the numerical range W(A).

Within the setting of bounded Hilbert space operators, we use extremal functions and vectors to show that several classical results on c-spectral sets fit into one abstract estimate. A key tool in this research is the double-layer potential, originally introduced in this context by Delyon–Delyon (1999). In light of this reference and more recent works, we examine the structure and relevance of the double-layer to the numerical range as a c-spectral set.

The discussion then moves to numerical ranges in general Banach algebras. Here the absence of an involution prevents the use of the double-layer potential techniques. Accordingly, we use alternative methods to provide several examples and counterexamples.

Finally, we look at the functional calculus of ρ-contractions on Hilbert spaces. We obtain a new sharp bound, refining an estimate by Ando–Okubo (1975) and extending a result by Drury (2008).