On an inequality of A.~Grothendieck concerning operators on $L^1$

Details On an inequality of A.~Grothendieck concerning operators on $L^1$ On an inequality of A.~Grothendieck concerning operators on $L^1$

1997-08-28

arxiv.org/abs/math/9708205v1

Mathematics

Functional Analysis

Scientific paper

In 1955, A.~Grothendieck proved a basic inequality which shows that any bounded linear operator between $L^1(\mu)$-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An elementary proof of this inequality is obtained via a new decomposition principle for the lattice of measurable functions. An exposition is also given of the M.~L\'evy extension theorem for operators defined on subspaces of $L^1(\mu)$-spaces.

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