CHAPTER EIGHT - OPERATORS BETWEEN L_p SPACES, 0 < p < 1

Summary

Introduction

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

From Theorem 7.12 of the previous chapter, any non-zero endomorphism of L_p, 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 l_p. We are now asserting considerably more. Our second assertion above trivially implies that a non-zero operator from L_p into L_p(μ) preserves a copy of since embeds isomorphically into L_p(A). Of course it also implies Pallaschke's original result that for 0 < p < 1, L_p admits no non-trivial compact endomorphisms.

Pallaschke's results on the endomorphisms of L_p, 0 < p < 1, appeared in 1973. A further step was taken by Berg- Peck-Porta [1973], who studied projections on L_0. Kwapien [1973] characterized completely the operators from L₀ to L₀(μ).

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