Vol. 53, No. 1, 1974

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The range of a contractive projection on an Lp-space

S. J. Bernau and Howard E. Lacey

Vol. 53 (1974), No. 1, 21–41

Suppose (X,Σ) is a measure space, 1 p < and P 2. Let Lp = Lp(X,Σ) be the usual space of equivalence classes of Σ-measurable functions f such that |f|p is integrable. A contractive projection on Lp is a linear operator P : Lp Lp such that P2 = P and P1. In this paper we give a complete description of such contractive projections in terms of conditional expectation operators. We also show that a closed subspace M of Lp is the range of a contractive projection if and only if M is isometrically isomorphic to another Lp-space. Another sufficient condition shows, in particular, that every closed vector sublattice of an Lp-space is the range of a positive contractive projection.

Mathematical Subject Classification 2000
Primary: 46E30
Received: 22 May 1973
Published: 1 July 1974
S. J. Bernau
Howard E. Lacey