On \({\mathcal L}_{p,\lambda}\) spaces for small \(\lambda\) (Q1066417)

This paper presents some results related to the problem of whether a \({\mathcal L}_{p,\lambda}\) subspace of \(L_ p\) for \(1\leq p\neq 2<\infty\) is close to \(L_ p(\mu)\) for some measure \(\mu\) provided that \(\lambda\) is close enough to 1. We show that there is a strong relationship between the answer to this for any one p and all others. In particular it is shown that if for some p there is a positive solution to the problem and the distance to \(L_ p(\mu)\) goes to one as \(\lambda\) goes to one then there is a positive solution for all p with similar control on the distance to \(L_ p(\mu)\). The proof depends heavily on the results of L. Dor and G. Schechtman for the finite dimensional case. We also prove some representation results for some special classes of operators on \(L_ 1\) and a factorization theorem (through \(L_ 1)\) for certain projections on \(L_ p\).