On a question of P.R. Chernoff and H.F. Trotter

TL;DR

This paper proves that any dissipative operator on a Banach space can be extended to generate a quasi-contractive semigroup, answering a question posed by Chernoff and Trotter.

Contribution

It establishes the existence of a quasi-dissipative extension for dissipative operators, enabling the generation of quasi-contractive semigroups in Banach spaces.

Findings

Dissipative operators can be extended to generate quasi-contractive semigroups.

The extension may require enlarging the Banach space.

Provides a positive resolution to Chernoff and Trotter's question.

Abstract

Let $A$ be a dissipative operator on a Banach space with a dense domain. It is proved that $A$ has a quasi-dissipative extension (possibly in an enlarged Banach space) which generates a quasi-contractive $C_0$-semigroup. \par This gives a positive answer to the question posed by P.R.Chernoff and H.F.Trotter.