Introduction

In this chapter we present some representation theorems for operators from Lp to Lp (μ), 0 < p < 1. The theorems have some important consequences; for example, we will show that a non-zero operator from Lp to Lp (μ), 0 < p < 1, is an isomorphism when restricted to Lp(A), for some set A of positive measure.

From Theorem 7.12 of the previous chapter, any non-zero endomorphism of Lp, 0 < p < 1, is an isomorphism on some infinite-dimensional subspace – and by Theorem 7.20 of the previous chapter, the subspace can be taken to be lp. We are now asserting considerably more. Our second assertion above trivially implies that a non-zero operator from Lp into Lp(μ) preserves a copy of since embeds isomorphically into Lp(A). Of course it also implies Pallaschke's original result that for 0 < p < 1, Lp admits no non-trivial compact endomorphisms.

Pallaschke's results on the endomorphisms of Lp, 0 < p < 1, appeared in 1973. A further step was taken by Berg- Peck-Porta [1973], who studied projections on L0. Kwapien [1973] characterized completely the operators from L0 to L0(μ).

Then Kalton [1978a] characterized completely the operators from Lp to Lp(μ)), 0 < p < 1, and derived a number of results on the structure of Lp, 0 < p < 1, including the above-mentioned one, as corollaries.