Theorems of Korovkin type for L_p-spaces

S. J. Bernau

Vol. 53 (1974), No. 1, 11–19

DOI: 10.2140/pjm.1974.53.11

Abstract

Suppose (X,Σ,μ) is a measure space, 1 < p < ∞,p≠2, and that (T_n) is a net of linear contractions on (real or complex) L_p(X,Σ,μ). Let M = {x ∈ L_p : T_nx → x} (M is the convergence set for (T_n)). It is obvious that M is a closed subspace of L_p; 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 L_p and hence is itself isometrically isomorphic to an L_p-space. If S ⊂ L_p(X,Σ,μ) we can define tke shadow, 𝒮(S) of S to be the set of all x in L_p such that T_nx → x for every net of linear contraclions (T_n) such that T_ny → y for all y ∈ S. We shall also give a complete description of 𝒮(S) (for p≠1,2,∞).

Mathematical Subject Classification 2000

Primary: 41A65

Milestones

Received: 21 June 1973

Published: 1 July 1974