Bounded functional calculi for unbounded operators

TL;DR

This paper reviews recent developments in bounded functional calculi for unbounded operators, extending classical theories to broader classes of functions and operators, with improved bounds and applications in operator theory.

Contribution

It introduces new bounded functional calculi for unbounded operators, extending existing theories and providing sharper bounds and broader applicability.

Findings

Extended Hille-Phillips calculus to new classes of functions

Provided sharper operator norm bounds

Unified various calculi under a common framework

Abstract

This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups, in particular for bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. They include functions outside the Hille-Phillips class, and they generally give sharper bounds for the norms of the resulting operators $f(A)$. The calculi are mostly based on appropriate reproducing formulas for the relevant classes of functions, and they rely on significant and interesting developments of function theory. They are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. They can also be used to derive several well-known operator norm-estimates, provide generalisations of some of them, and extend the general theory of operator semigroups. Our aim is to help readers to make use of these calculi without having to understand the details of their construction.