Functional calculi for sectorial operators and related function theory

TL;DR

This paper develops advanced bounded functional calculi for sectorial operators on Banach spaces, extending existing theories with new function classes, sharper estimates, and sectoriality adaptations, leading to improved operator norm estimates.

Contribution

It introduces two new bounded functional calculi for sectorial operators, enhancing the analytic Besov function calculus and generalizing key estimates.

Findings

Enhanced functional calculi compatible with standard theories

Generalized and sharpened operator norm estimates

Development of associated function spaces and spectral theorems

Abstract

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalizing and sharpening estimates, and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways which are interesting and significant. As consequences of our calculi, we derive several well-known operator norm-estimates and provide generalizations of some of them.