Positive Operators on L p-spaces

Throughout this paper we denote by

L

p

the Banach lattice of

p

-integrable functions on a

σ

-finite measure space (

X, B, μ

), where 1 ≤

p

≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on

L

p

-spaces. For general information about positive operators on Banach lattices we refer to the texts [

1

]. [

20

], and [

36

]. Our focus on

L

p

-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of

L

p

-spaces. We will discuss results related to the boundedness of positive linear operators on

L

p

-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from

L

p

to

L

q

. Then we will show how this result relates to the change of density result of Weis [

33

]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on

L

p

-spaces and discuss as an example the weighted composition operators on

L

p

-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikišin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.