The group reduction for bounded cosine functions on UMD spaces

TL;DR

This paper proves that on UMD spaces, a bounded cosine operator function generated by A implies that i(-A)^{1/2} generates a bounded C_0-group, using a transference principle.

Contribution

It introduces a transference principle for cosine functions to establish a new connection between cosine functions and C_0-groups on UMD spaces.

Findings

Bounded cosine functions imply bounded C_0-groups via fractional powers.

The proof employs a transference principle specific to UMD spaces.

The result extends the understanding of operator functions on Banach spaces.

Abstract

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(-A)^{1/2} generates a bounded C_0-group. The proof uses a transference principle for cosine functions.