Suppose (X,Σ,μ) is a
measure space, 1 < p < ∞,p≠2, and that (Tn) is a net of linear contractions on (real
or complex) Lp(X,Σ,μ). Let M = {x ∈ Lp: Tnx → x} (M is the convergence set for
(Tn)). It is obvious that M is a closed subspace of Lp; indeed this would be true for
an arbitrary normed space. In this paper we shall show that M is the range of a
contractive projection on Lp and hence is itself isometrically isomorphic to an
Lp-space. If S ⊂ Lp(X,Σ,μ) we can define tke shadow, 𝒮(S) of S to be the set of all
x in Lp such that Tnx → x for every net of linear contraclions (Tn) such that
Tny → y for all y ∈ S. We shall also give a complete description of 𝒮(S) (for
p≠1,2,∞).
