On almost everywhere convergence of Abel means of contraction semigroups

Stephen Allan McGrath

Vol. 65 (1976), No. 2, 405–408

DOI: 10.2140/pjm.1976.65.405

Abstract

Let (X,Σ,μ) be a σ-finite measure space and L_p(X,Σ,μ), 1 ≦ p ≦∞, the usual Banach spaces of complex valued functions. Let {T_t : t ≧ 0} be a strongly continuous semigroup of contractions of L_p(X,Σ,μ) for some 1 ≦ p < ∞ and set R_λf = ∫ ₀^∞e^−λtT_tf dt, λ > 0. If ∥T_t∥_∞ ≦ 1 for all t ≧ 0, then lim_λ→∞λR_λf(x) = f(x) a.e. for all f ∈ L_p(X,Σ,μ).

Mathematical Subject Classification 2000

Primary: 47D05, 47D05

Secondary: 47A20

Milestones

Received: 14 April 1976

Revised: 30 June 1976

Published: 1 August 1976