A transference principle for general groups and functional calculus on UMD spaces

TL;DR

This paper establishes a transference principle for strongly continuous groups on Banach spaces with the UMD property, enabling new estimates for functional calculus of the group generator, with implications for sectorial operators.

Contribution

It introduces a transference principle applicable to general groups on UMD Banach spaces, extending classical results to broader settings and deriving consequences for sectorial operators.

Findings

Transference principle for strongly continuous groups on Banach spaces.

Estimates for functional calculus of the group generator in UMD spaces.

Bounded H-infinity calculus for generators of cosine functions on UMD spaces.

Abstract

We prove a transference principle for general (i.e., not necessarily bounded) strongly continuous groups on Banach spaces. If the Banach space has the UMD property, the transference principle leads to estimates for the functional calculus of the group generator. In the Hilbert space case, the results cover classical theorems of McIntosh and Boyadzhiev-de Laubenfels; in the UMD case they are analogues of classical results by Hieber and Pruess. By using functional calculus methods, consequences for sectorial operators are derived. For instance it is proved, that every generator of a cosine function on a UMD space has bounded H-infinity calculus on sectors.