Contraction semigroups in Lebesgue space

Ryōtarō Satō

Vol. 78 (1978), No. 1, 251–259

DOI: 10.2140/pjm.1978.78.251

Abstract

Let (T_t : t > 0) be a strongly continuous semigroup of linear contractions on L₁(X,Σ,μ), where (X,Σ,μ) is a σ-finite measure space. Without assuming the initial continuity of the semigroup it is shown that (T_t : t > 0) is dominated by a strongly continuous semigroup (S_t : t > 0) of positive linear contractions on L₁(X,Σ,μ), i.e., that |T_tf|≦ S_t|f| holds a.e. on X for all f ∈ L₁(X,Σ,μ) and all t > 0. As an application, a representation of (T_t : t > 0) in terms of (S_t : t > 0) is obtained, and the question of the almost everywhere convergence of 1∕b∫ ₀^bT_tf dt as b → +0 is considered.

Mathematical Subject Classification

Primary: 47D05, 47D05

Milestones

Received: 27 July 1977

Published: 1 September 1978