OPERATORS BETWEEN Lp SPACES, 0 < p < 1
N. J. Kalton, N. Tenney Peck, James W. Roberts · Cambridge University Press eBooks · 1984
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 L 0 to L 0 (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.