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

M. A. Perelmuter

TL;DR

The paper answers Chernoff and Trotter's question by showing that a closed densely defined dissipative operator $A$ on a Banach space can be extended to a quasi-dissipative operator $B$ on an enlarged space that generates a $C_0$-semigroup. The construction uses an enlarged space $\mathcal{E}=\ell^1$-sum of countably many copies of the original space, with $B$ extending $A$ and satisfying ${\rm Ran}(\lambda-B)=\mathcal{E}$ for some $\lambda>0$, yielding a quasi-contractive semigroup with $\|e^{tB}\|\le e^{t}$. A simple modification allows achieving $\|e^{tB}\|\le e^{\varepsilon t}$ for any $\varepsilon>0$, approaching contraction. This provides a concrete, constructive positive answer to Chernoff–Trotter and extends the Lumer–Phillips framework to quasi-dissipative extensions in enlarged spaces.

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.

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

TL;DR

The paper answers Chernoff and Trotter's question by showing that a closed densely defined dissipative operator on a Banach space can be extended to a quasi-dissipative operator on an enlarged space that generates a -semigroup. The construction uses an enlarged space -sum of countably many copies of the original space, with extending and satisfying for some , yielding a quasi-contractive semigroup with . A simple modification allows achieving for any , approaching contraction. This provides a concrete, constructive positive answer to Chernoff–Trotter and extends the Lumer–Phillips framework to quasi-dissipative extensions in enlarged spaces.

Abstract

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

Paper Structure

This paper contains 2 sections, 1 theorem, 4 equations.

Table of Contents

  1. Introduction
  2. Main Result

Key Result

Theorem 1

Let $A$ be a dissipative operator with dense domain in a Banach space $\mathcal{B}$. Then $A$ has a quasi-dissipative extension $A\subset B$ (possibly in an enlarged Banach space) which is a generator of the $C_0$-semigroup $e^{tB}$ and $\left\|e^{tB}\right\|\leq e^t$.

Theorems & Definitions (2)

  • Theorem 1
  • proof