Concerning the infinite differentiability of semigroup motions

J. W. Spellmann

Vol. 30 (1969), No. 2, 519–523

DOI: 10.2140/pjm.1969.30.519

Abstract

Let S be a real Banach space. Let C denote the infinitesimal generator of a strongly continuous semigroup T of bounded linear transformations on S. This paper presents a construction which proves that for each b > 1 there is a dense subset D(b) of S so that if p is in D(b), then

(A) p is in the domain of Cⁿ for all positive integers n and

(B) lim_n→∞∥Cⁿp∥(n!)^−b = 0.

Condition (B) will be used in §3 to obtain series solutions to the partial differential equations U₁₂ = CU and U₁₁ = CU.

Mathematical Subject Classification

Primary: 47.50

Milestones

Received: 1 August 1968

Published: 1 August 1969