Transference Principles for Semigroups and a Theorem of Peller

TL;DR

This paper develops a unified transference framework for operator semigroups, extending classical results and deriving new functional calculus estimates, including a novel proof of Peller's 1982 theorem and generalizations to Banach spaces.

Contribution

It introduces a new transference principle applicable to non-group semigroups, enabling broader functional calculus estimates and extending classical results to Banach spaces.

Findings

Unified transference principles for operator semigroups.

New proof of Peller's classical results from 1982.

Extension of results to Banach spaces and applications to singular integrals.

Abstract

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows to recover the classical transference results of Calder\'on, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, culminating in a new proof of classical results by Peller from 1982. The method allows a generalization of his results away from Hilbert spaces to $\Ell{p}$-spaces and --- involving the concept of $\gamma$-boundedness --- to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.