Vol. 78, No. 1, 1978

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Contraction semigroups in Lebesgue space

Ryōtarō Satō

Vol. 78 (1978), No. 1, 251–259
Abstract

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

Mathematical Subject Classification
Primary: 47D05, 47D05
Milestones
Received: 27 July 1977
Published: 1 September 1978
Authors
Ryōtarō Satō